Theorem cnvimassrndm 6116

Description: The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 6048 for subsets. (Contributed by AV, 18-Sep-2024.)

Assertion

Ref	Expression
cnvimassrndm	⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)

Proof of Theorem cnvimassrndm

Step	Hyp	Ref	Expression
1		ssequn1 4126	. 2 ⊢ (ran 𝐹 ⊆ 𝐴 ↔ (ran 𝐹 ∪ 𝐴) = 𝐴)
2		imaeq2 6021	. . . . 5 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (ran 𝐹 ∪ 𝐴)))
3		imaundi 6113	. . . . 5 ⊢ (◡𝐹 “ (ran 𝐹 ∪ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴))
4	2, 3	eqtrdi 2787	. . . 4 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)))
5		cnvimarndm 6048	. . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
6	5	uneq1i 4104	. . . . 5 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = (dom 𝐹 ∪ (◡𝐹 “ 𝐴))
7		cnvimass 6047	. . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
8		ssequn2 4129	. . . . . 6 ⊢ ((◡𝐹 “ 𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹)
9	7, 8	mpbi 230	. . . . 5 ⊢ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹
10	6, 9	eqtri 2759	. . . 4 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = dom 𝐹
11	4, 10	eqtrdi 2787	. . 3 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = dom 𝐹)
12	11	eqcoms 2744	. 2 ⊢ ((ran 𝐹 ∪ 𝐴) = 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
13	1, 12	sylbi 217	1 ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)

Syntax hints:   → wi 4   = wceq 1542   ∪ cun 3887   ⊆ wss 3889  ^◡ccnv 5630  dom cdm 5631  ran crn 5632   “ cima 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  fnco  6616  fimacnv  6690