Improve _element_constructor_ of image set

src/sage/sets/image_set.py

@@ -53,7 +53,9 @@ class ImageSubobject(Parent):
       - ``True``: ``map`` is known to be injective
       - ``'check'``: raise an error when ``map`` is not injective
 
-    - ``inverse`` -- a function (optional); a map from `f(X)` to `X`
+    - ``inverse`` -- a function (optional); a map from `f(X)` to `X`.
+      It is not recommended to provide this, if ``map.inverse_image`` exists,
+      it will be used automatically.
 
     EXAMPLES::
 
@@ -80,6 +82,17 @@ def __init__(self, map, domain_subset, *, category=None, is_injective=None, inve
             sage: Im = f.image()
             sage: TestSuite(Im).run(skip=['_test_an_element', '_test_pickling',
             ....:                         '_test_some_elements', '_test_elements'])
+
+        TESTS:
+
+        Implementing ``inverse_image`` automatically makes :meth:`__contains__` work::
+
+            sage: R.<x> = QQ[]
+            sage: S.<y> = QQ[]
+            sage: R.hom([y^2]).inverse_image(y^4)
+            x^2
+            sage: y^4 in R.hom([y^2]).image()
+            True
         """
         if not isinstance(domain_subset, Parent):
             from sage.sets.set import Set
@@ -127,6 +140,11 @@ def map(arg):
         Parent.__init__(self, category=category)
 
         self._map = map
+        if inverse is None:
+            try:
+                inverse = map.inverse_image
+            except AttributeError:
+                pass
         self._inverse = inverse
         self._domain_subset = domain_subset
         self._is_injective = is_injective