Theorem afv2eq12d 47678

Description: Equality deduction for function value, analogous to fveq12d 6834. (Contributed by AV, 4-Sep-2022.)

Hypotheses

Ref	Expression
afv2eq12d.1	⊢ (𝜑 → 𝐹 = 𝐺)
afv2eq12d.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
afv2eq12d	⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Proof of Theorem afv2eq12d

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		afv2eq12d.1	. . . 4 ⊢ (𝜑 → 𝐹 = 𝐺)
2		afv2eq12d.2	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
3	1, 2	dfateq12d 47589	. . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵))
4		eqidd 2740	. . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥)
5	2, 1, 4	breq123d 5086	. . . 4 ⊢ (𝜑 → (𝐴𝐹𝑥 ↔ 𝐵𝐺𝑥))
6	5	iotabidv 6469	. . 3 ⊢ (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
7	1	rneqd 5880	. . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐺)
8	7	unieqd 4851	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺)
9	8	pweqd 4546	. . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺)
10	3, 6, 9	ifbieq12d 4483	. 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺))
11		df-afv2 47672	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
12		df-afv2 47672	. 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)
13	10, 11, 12	3eqtr4g 2799	1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Syntax hints:   → wi 4   = wceq 1547  ifcif 4454  𝒫 cpw 4529  ∪ cuni 4838   class class class wbr 5072  ran crn 5619  ℩cio 6439   defAt wdfat 47579  ''''cafv2 47671

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

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-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-iota 6441  df-fun 6487  df-dfat 47582  df-afv2 47672

This theorem is referenced by:  afv2eq1  47679  afv2eq2  47680  csbafv212g  47682