Theorem cnvimassrndm 6103

Description: The preimage of a superset of the range of a class is the domain of the class. Generalization of cnvimarndm 6035 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 4115	. 2 ⊢ (ran 𝐹 ⊆ 𝐴 ↔ (ran 𝐹 ∪ 𝐴) = 𝐴)
2		imaeq2 6008	. . . . 5 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (ran 𝐹 ∪ 𝐴)))
3		imaundi 6100	. . . . 5 ⊢ (◡𝐹 “ (ran 𝐹 ∪ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴))
4	2, 3	eqtrdi 2790	. . . 4 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)))
5		cnvimarndm 6035	. . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
6	5	uneq1i 4094	. . . . 5 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = (dom 𝐹 ∪ (◡𝐹 “ 𝐴))
7		cnvimass 6034	. . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
8		ssequn2 4118	. . . . . 6 ⊢ ((◡𝐹 “ 𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹)
9	7, 8	mpbi 231	. . . . 5 ⊢ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹
10	6, 9	eqtri 2762	. . . 4 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = dom 𝐹
11	4, 10	eqtrdi 2790	. . 3 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = dom 𝐹)
12	11	eqcoms 2747	. 2 ⊢ ((ran 𝐹 ∪ 𝐴) = 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
13	1, 12	sylbi 218	1 ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)

Syntax hints:   → wi 4   = wceq 1547   ∪ cun 3881   ⊆ wss 3883  ^◡ccnv 5617  dom cdm 5618  ran crn 5619   “ cima 5621

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-xp 5624  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631

This theorem is referenced by:  fnco  6603  fimacnv  6677