Demostrar que<\/p>\n
\n f[s] \u2286 u \u2194 s \u2286 f\u207b\u00b9[u]\n<\/pre>\nPara ello, completar la siguiente teor\u00eda de Lean:<\/p>\n
\nimport data.set.basic\n\nopen set\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\nvariable f : \u03b1 \u2192 \u03b2\nvariable s : set \u03b1\nvariable u : set \u03b2\n\nexample : f '' s \u2286 u \u2194 s \u2286 f \u207b\u00b9' u :=\nsorry\n<\/pre>\n[expand title=\u00bbSoluciones con Lean\u00bb]<\/p>\n
\r\nimport data.set.basic\r\n\r\nopen set\r\n\r\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\r\nvariable f : \u03b1 \u2192 \u03b2\r\nvariable s : set \u03b1\r\nvariable u : set \u03b2\r\n\r\n-- 1\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' s \u2286 u \u2194 s \u2286 f \u207b\u00b9' u :=\r\nbegin\r\n split,\r\n { intros h x xs,\r\n apply mem_preimage.mpr,\r\n apply h,\r\n apply mem_image_of_mem,\r\n exact xs, },\r\n { intros h y hy,\r\n rcases hy with \u27e8x, xs, fxy\u27e9,\r\n rw \u2190 fxy,\r\n exact h xs, },\r\nend\r\n\r\n-- 2\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' s \u2286 u \u2194 s \u2286 f \u207b\u00b9' u :=\r\nbegin\r\n split,\r\n { intros h x xs,\r\n apply h,\r\n apply mem_image_of_mem,\r\n exact xs, },\r\n { rintros h y \u27e8x, xs, rfl\u27e9,\r\n exact h xs, },\r\nend\r\n\r\n-- 3\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' s \u2286 u \u2194 s \u2286 f \u207b\u00b9' u :=\r\nimage_subset_iff\r\n\r\n-- 4\u00aa demostraci\u00f3n\r\n-- ===============\r\n\r\nexample : f '' s \u2286 u \u2194 s \u2286 f \u207b\u00b9' u :=\r\nby simp\r\n<\/pre>\n