diff --git a/hypothesis-python/src/hypothesis/_strategies.py b/hypothesis-python/src/hypothesis/_strategies.py
index 97d0c7c461..8272fe3015 100644
--- a/hypothesis-python/src/hypothesis/_strategies.py
+++ b/hypothesis-python/src/hypothesis/_strategies.py
@@ -87,6 +87,7 @@
     ListStrategy,
     TupleStrategy,
     UniqueListStrategy,
+    UniqueSampledListStrategy,
 )
 from hypothesis.searchstrategy.datetime import (
     DateStrategy,
@@ -775,6 +776,26 @@ def unique_by(x):
         for i, f in enumerate(unique_by):
             if not callable(f):
                 raise InvalidArgument("unique_by[%i]=%r is not a callable" % (i, f))
+        # Note that lazy strategies automatically unwrap when passed to a defines_strategy
+        # function.
+        if isinstance(elements, SampledFromStrategy):
+            element_count = len(elements.elements)
+            if min_size > element_count:
+                raise InvalidArgument(
+                    "Cannot create a collection of min_size=%r unique elements with "
+                    "values drawn from only %d distinct elements"
+                    % (min_size, element_count)
+                )
+
+            if max_size is not None:
+                max_size = min(max_size, element_count)
+            else:
+                max_size = element_count
+
+            return UniqueSampledListStrategy(
+                elements=elements, max_size=max_size, min_size=min_size, keys=unique_by
+            )
+
         return UniqueListStrategy(
             elements=elements, max_size=max_size, min_size=min_size, keys=unique_by
         )
diff --git a/hypothesis-python/src/hypothesis/internal/conjecture/junkdrawer.py b/hypothesis-python/src/hypothesis/internal/conjecture/junkdrawer.py
index 13185ae216..77b5b1c184 100644
--- a/hypothesis-python/src/hypothesis/internal/conjecture/junkdrawer.py
+++ b/hypothesis-python/src/hypothesis/internal/conjecture/junkdrawer.py
@@ -173,3 +173,51 @@ def binary_search(lo, hi, f):
 def uniform(random, n):
     """Returns an hbytes of length n, distributed uniformly at random."""
     return int_to_bytes(random.getrandbits(n * 8), n)
+
+
+class LazySequenceCopy(object):
+    """A "copy" of a sequence that works by inserting a mask in front
+    of the underlying sequence, so that you can mutate it without changing
+    the underlying sequence. Effectively behaves as if you could do list(x)
+    in O(1) time. The full list API is not supported yet but there's no reason
+    in principle it couldn't be."""
+
+    def __init__(self, values):
+        self.__values = values
+        self.__len = len(values)
+        self.__mask = None
+
+    def __len__(self):
+        return self.__len
+
+    def pop(self):
+        if len(self) == 0:
+            raise IndexError("Cannot pop from empty list")
+        result = self[-1]
+        self.__len -= 1
+        if self.__mask is not None:
+            self.__mask.pop(self.__len, None)
+        return result
+
+    def __getitem__(self, i):
+        i = self.__check_index(i)
+        default = self.__values[i]
+        if self.__mask is None:
+            return default
+        else:
+            return self.__mask.get(i, default)
+
+    def __setitem__(self, i, v):
+        i = self.__check_index(i)
+        if self.__mask is None:
+            self.__mask = {}
+        self.__mask[i] = v
+
+    def __check_index(self, i):
+        n = len(self)
+        if i < -n or i >= n:
+            raise IndexError("Index %d out of range [0, %d)" % (i, n))
+        if i < 0:
+            i += n
+        assert 0 <= i < n
+        return i
diff --git a/hypothesis-python/src/hypothesis/searchstrategy/collections.py b/hypothesis-python/src/hypothesis/searchstrategy/collections.py
index 98c9159810..d0f6c45dc1 100644
--- a/hypothesis-python/src/hypothesis/searchstrategy/collections.py
+++ b/hypothesis-python/src/hypothesis/searchstrategy/collections.py
@@ -20,6 +20,7 @@
 import hypothesis.internal.conjecture.utils as cu
 from hypothesis.errors import InvalidArgument
 from hypothesis.internal.compat import OrderedDict
+from hypothesis.internal.conjecture.junkdrawer import LazySequenceCopy
 from hypothesis.internal.conjecture.utils import combine_labels
 from hypothesis.searchstrategy.strategies import (
     MappedSearchStrategy,
@@ -169,6 +170,40 @@ def do_draw(self, data):
         return result
 
 
+class UniqueSampledListStrategy(ListStrategy):
+    def __init__(self, elements, min_size, max_size, keys):
+        super(UniqueSampledListStrategy, self).__init__(elements, min_size, max_size)
+        self.keys = keys
+
+    def do_draw(self, data):
+        should_draw = cu.many(
+            data,
+            min_size=self.min_size,
+            max_size=self.max_size,
+            average_size=self.average_size,
+        )
+        seen_sets = tuple(set() for _ in self.keys)
+        result = []
+
+        remaining = LazySequenceCopy(self.element_strategy.elements)
+
+        while should_draw.more():
+            i = len(remaining) - 1
+            j = cu.integer_range(data, 0, i)
+            if j != i:
+                remaining[i], remaining[j] = remaining[j], remaining[i]
+            value = remaining.pop()
+
+            if all(key(value) not in seen for (key, seen) in zip(self.keys, seen_sets)):
+                for key, seen in zip(self.keys, seen_sets):
+                    seen.add(key(value))
+                result.append(value)
+            else:
+                should_draw.reject()
+        assert self.max_size >= len(result) >= self.min_size
+        return result
+
+
 class FixedKeysDictStrategy(MappedSearchStrategy):
     """A strategy which produces dicts with a fixed set of keys, given a
     strategy for each of their equivalent values.
diff --git a/hypothesis-python/src/hypothesis/searchstrategy/lazy.py b/hypothesis-python/src/hypothesis/searchstrategy/lazy.py
index ed57e303dc..5b261caf55 100644
--- a/hypothesis-python/src/hypothesis/searchstrategy/lazy.py
+++ b/hypothesis-python/src/hypothesis/searchstrategy/lazy.py
@@ -80,7 +80,7 @@ def __init__(self, function, args, kwargs):
         SearchStrategy.__init__(self)
         self.__wrapped_strategy = None
         self.__representation = None
-        self.__function = function
+        self.function = function
         self.__args = args
         self.__kwargs = kwargs
 
@@ -109,11 +109,11 @@ def wrapped_strategy(self):
                 k: unwrap_strategies(v) for k, v in self.__kwargs.items()
             }
 
-            base = self.__function(*self.__args, **self.__kwargs)
+            base = self.function(*self.__args, **self.__kwargs)
             if unwrapped_args == self.__args and unwrapped_kwargs == self.__kwargs:
                 self.__wrapped_strategy = base
             else:
-                self.__wrapped_strategy = self.__function(
+                self.__wrapped_strategy = self.function(
                     *unwrapped_args, **unwrapped_kwargs
                 )
         return self.__wrapped_strategy
@@ -127,7 +127,7 @@ def __repr__(self):
         if self.__representation is None:
             _args = self.__args
             _kwargs = self.__kwargs
-            argspec = getfullargspec(self.__function)
+            argspec = getfullargspec(self.function)
             defaults = dict(argspec.kwonlydefaults or {})
             if argspec.defaults is not None:
                 for name, value in zip(
@@ -136,19 +136,19 @@ def __repr__(self):
                     defaults[name] = value
             if len(argspec.args) > 1 or argspec.defaults:
                 _args, _kwargs = convert_positional_arguments(
-                    self.__function, _args, _kwargs
+                    self.function, _args, _kwargs
                 )
             else:
                 _args, _kwargs = convert_keyword_arguments(
-                    self.__function, _args, _kwargs
+                    self.function, _args, _kwargs
                 )
             kwargs_for_repr = dict(_kwargs)
             for k, v in defaults.items():
                 if k in kwargs_for_repr and kwargs_for_repr[k] is defaults[k]:
                     del kwargs_for_repr[k]
             self.__representation = "%s(%s)" % (
-                self.__function.__name__,
-                arg_string(self.__function, _args, kwargs_for_repr, reorder=False),
+                self.function.__name__,
+                arg_string(self.function, _args, kwargs_for_repr, reorder=False),
             )
         return self.__representation
 
diff --git a/hypothesis-python/tests/cover/test_conjecture_junkdrawer.py b/hypothesis-python/tests/cover/test_conjecture_junkdrawer.py
new file mode 100644
index 0000000000..21a3bff8c5
--- /dev/null
+++ b/hypothesis-python/tests/cover/test_conjecture_junkdrawer.py
@@ -0,0 +1,56 @@
+# coding=utf-8
+#
+# This file is part of Hypothesis, which may be found at
+# https://github.com/HypothesisWorks/hypothesis/
+#
+# Most of this work is copyright (C) 2013-2019 David R. MacIver
+# (david@drmaciver.com), but it contains contributions by others. See
+# CONTRIBUTING.rst for a full list of people who may hold copyright, and
+# consult the git log if you need to determine who owns an individual
+# contribution.
+#
+# This Source Code Form is subject to the terms of the Mozilla Public License,
+# v. 2.0. If a copy of the MPL was not distributed with this file, You can
+# obtain one at https://mozilla.org/MPL/2.0/.
+#
+# END HEADER
+
+from __future__ import absolute_import, division, print_function
+
+import pytest
+
+from hypothesis.internal.conjecture.junkdrawer import LazySequenceCopy
+
+
+def test_out_of_range():
+    x = LazySequenceCopy([1, 2, 3])
+
+    with pytest.raises(IndexError):
+        x[3]
+
+    with pytest.raises(IndexError):
+        x[-4]
+
+
+def test_pass_through():
+    x = LazySequenceCopy([1, 2, 3])
+    assert x[0] == 1
+    assert x[1] == 2
+    assert x[2] == 3
+
+
+def test_can_assign_without_changing_underlying():
+    underlying = [1, 2, 3]
+    x = LazySequenceCopy(underlying)
+    x[1] = 10
+    assert x[1] == 10
+    assert underlying[1] == 2
+
+
+def test_pop():
+    x = LazySequenceCopy([2, 3])
+    assert x.pop() == 3
+    assert x.pop() == 2
+
+    with pytest.raises(IndexError):
+        x.pop()
diff --git a/hypothesis-python/tests/cover/test_direct_strategies.py b/hypothesis-python/tests/cover/test_direct_strategies.py
index 54d2572406..37a7cd59ca 100644
--- a/hypothesis-python/tests/cover/test_direct_strategies.py
+++ b/hypothesis-python/tests/cover/test_direct_strategies.py
@@ -119,6 +119,7 @@ def fn_ktest(*fnkwargs):
     (ds.lists, {"elements": ds.integers(), "unique_by": 1}),
     (ds.lists, {"elements": ds.integers(), "unique_by": ()}),
     (ds.lists, {"elements": ds.integers(), "unique_by": (1,)}),
+    (ds.lists, {"elements": ds.sampled_from([0, 1]), "min_size": 3, "unique": True}),
     (ds.text, {"min_size": 10, "max_size": 9}),
     (ds.text, {"alphabet": [1]}),
     (ds.text, {"alphabet": ["abc"]}),
diff --git a/hypothesis-python/tests/cover/test_sampled_from.py b/hypothesis-python/tests/cover/test_sampled_from.py
index ce8a3884b6..d7e887791b 100644
--- a/hypothesis-python/tests/cover/test_sampled_from.py
+++ b/hypothesis-python/tests/cover/test_sampled_from.py
@@ -20,6 +20,7 @@
 import collections
 import enum
 
+import hypothesis.strategies as st
 from hypothesis import given
 from hypothesis.errors import FailedHealthCheck, InvalidArgument, Unsatisfiable
 from hypothesis.internal.compat import hrange
@@ -76,3 +77,18 @@ def test_easy_filtered_sampling():
 @given(sampled_from(hrange(100)).filter(lambda x: x == 99))
 def test_filtered_sampling_finds_rare_value(x):
     assert x == 99
+
+
+@given(st.sets(st.sampled_from(range(50)), min_size=50))
+def test_efficient_sets_of_samples(x):
+    assert x == set(range(50))
+
+
+@given(st.lists(st.sampled_from([0] * 100), unique=True))
+def test_does_not_include_duplicates_even_when_duplicated_in_collection(ls):
+    assert len(ls) <= 1
+
+
+@given(st.lists(st.sampled_from(hrange(100)), max_size=3, unique=True))
+def test_max_size_is_respected_with_unique_sampled_from(ls):
+    assert len(ls) <= 3
diff --git a/hypothesis-python/tests/nocover/test_sampled_from.py b/hypothesis-python/tests/nocover/test_sampled_from.py
index ab6a8c0413..85af3db7e9 100644
--- a/hypothesis-python/tests/nocover/test_sampled_from.py
+++ b/hypothesis-python/tests/nocover/test_sampled_from.py
@@ -21,7 +21,7 @@
 
 import hypothesis.strategies as st
 from hypothesis import given
-from hypothesis.errors import FailedHealthCheck
+from hypothesis.errors import InvalidArgument
 from hypothesis.internal.compat import hrange
 from tests.common.utils import counts_calls, fails_with
 
@@ -64,7 +64,7 @@ def test_chained_filters_find_rare_value(x):
     assert x == 80
 
 
-@fails_with(FailedHealthCheck)
+@fails_with(InvalidArgument)
 @given(st.sets(st.sampled_from(range(10)), min_size=11))
 def test_unsat_sets_of_samples(x):
     assert False