Theorem cnvimainrn 6994

Description: The preimage of the intersection of the range of a class and a class 𝐴 is the preimage of the class 𝐴. (Contributed by AV, 17-Sep-2024.)

Assertion

Ref	Expression
cnvimainrn	⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴))

Proof of Theorem cnvimainrn

Step	Hyp	Ref	Expression
1		inpreima 6991	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴)))
2		cnvimass 6013	. . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
3		cnvimarndm 6014	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
4	2, 3	sseqtrri 3968	. . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹)
5		df-ss 3914	. . . 4 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴))
6	4, 5	mpbi 229	. . 3 ⊢ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴)
7	6	ineqcomi 4149	. 2 ⊢ ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴)) = (◡𝐹 “ 𝐴)
8	1, 7	eqtrdi 2792	1 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∩ cin 3896   ⊆ wss 3897  ^◡ccnv 5613  dom cdm 5614  ran crn 5615   “ cima 5617  Fun wfun 6467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-br 5090  df-opab 5152  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6475

This theorem is referenced by:  fcoreslem1  44897