Theorem cnvimainrn 7010

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 7007	. 2 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴)))
2		cnvimass 6039	. . . . 5 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
3		cnvimarndm 6040	. . . . 5 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
4	2, 3	sseqtrri 3981	. . . 4 ⊢ (◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹)
5		dfss2 3917	. . . 4 ⊢ ((◡𝐹 “ 𝐴) ⊆ (◡𝐹 “ ran 𝐹) ↔ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴))
6	4, 5	mpbi 230	. . 3 ⊢ ((◡𝐹 “ 𝐴) ∩ (◡𝐹 “ ran 𝐹)) = (◡𝐹 “ 𝐴)
7	6	ineqcomi 4161	. 2 ⊢ ((◡𝐹 “ ran 𝐹) ∩ (◡𝐹 “ 𝐴)) = (◡𝐹 “ 𝐴)
8	1, 7	eqtrdi 2785	1 ⊢ (Fun 𝐹 → (◡𝐹 “ (ran 𝐹 ∩ 𝐴)) = (◡𝐹 “ 𝐴))

Syntax hints:   → wi 4   = wceq 1541   ∩ cin 3898   ⊆ wss 3899  ^◡ccnv 5621  dom cdm 5622  ran crn 5623   “ cima 5625  Fun wfun 6484

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-fun 6492

This theorem is referenced by:  fcoreslem1  47251