Theorem cnvimassrndmOLD 6218

Description: Obsolete version of cnvimarndm 6098 as of 29-Sep-2026. (Contributed by AV, 18-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem cnvimassrndmOLD

Step	Hyp	Ref	Expression
1		ssequn1 4187	. 2 ⊢ (ran 𝐹 ⊆ 𝐴 ↔ (ran 𝐹 ∪ 𝐴) = 𝐴)
2		imaeq2 6066	. . . . 5 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = (◡𝐹 “ (ran 𝐹 ∪ 𝐴)))
3		imaundi 6159	. . . . 5 ⊢ (◡𝐹 “ (ran 𝐹 ∪ 𝐴)) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴))
4	2, 3	eqtrdi 2814	. . . 4 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)))
5		cnvimarndm 6098	. . . . . 6 ⊢ (◡𝐹 “ ran 𝐹) = dom 𝐹
6	5	uneq1i 4166	. . . . 5 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = (dom 𝐹 ∪ (◡𝐹 “ 𝐴))
7		cnvimass 6217	. . . . . 6 ⊢ (◡𝐹 “ 𝐴) ⊆ dom 𝐹
8		ssequn2 4190	. . . . . 6 ⊢ ((◡𝐹 “ 𝐴) ⊆ dom 𝐹 ↔ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹)
9	7, 8	mpbi 233	. . . . 5 ⊢ (dom 𝐹 ∪ (◡𝐹 “ 𝐴)) = dom 𝐹
10	6, 9	eqtri 2786	. . . 4 ⊢ ((◡𝐹 “ ran 𝐹) ∪ (◡𝐹 “ 𝐴)) = dom 𝐹
11	4, 10	eqtrdi 2814	. . 3 ⊢ (𝐴 = (ran 𝐹 ∪ 𝐴) → (◡𝐹 “ 𝐴) = dom 𝐹)
12	11	eqcoms 2771	. 2 ⊢ ((ran 𝐹 ∪ 𝐴) = 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)
13	1, 12	sylbi 220	1 ⊢ (ran 𝐹 ⊆ 𝐴 → (◡𝐹 “ 𝐴) = dom 𝐹)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900   ∪ cun 4152  ^◡ccnv 5657  dom cdm 5658  ran crn 5659   “ cima 5661

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-cnv 5666  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671

This theorem is used by: (None)