Theorem cnvimainrn 7074

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 7071	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴)))
2		cnvimass 6204	. . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
3		cnvimarndm 6085	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
4	2, 3	sseqtrri 3961	. . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹)
5		dfss2 4096	. . . 4 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴))
6	4, 5	mpbi 233	. . 3 ⊢ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴)
7	6	ineqcomi 4110	. 2 ⊢ ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴)) = (◡𝐹 “ 𝐴)
8	1, 7	eqtrdi 2814	1 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ⊆ wss 3899   ∩ cin 4091  ^◡ccnv 5653  dom cdm 5654  ran crn 5655   “ cima 5657  Fun wfun 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-12 2215  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-id 5549  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6548

This theorem is used by:  fcoreslem1  48272