Theorem afv2eq12d 47457

Description: Equality deduction for function value, analogous to fveq12d 6841. (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 47368	. . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵))
4		eqidd 2737	. . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥)
5	2, 1, 4	breq123d 5112	. . . 4 ⊢ (𝜑 → (𝐴𝐹𝑥 ↔ 𝐵𝐺𝑥))
6	5	iotabidv 6476	. . 3 ⊢ (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
7	1	rneqd 5887	. . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐺)
8	7	unieqd 4876	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺)
9	8	pweqd 4571	. . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺)
10	3, 6, 9	ifbieq12d 4508	. 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺))
11		df-afv2 47451	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
12		df-afv2 47451	. 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)
13	10, 11, 12	3eqtr4g 2796	1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Syntax hints:   → wi 4   = wceq 1541  ifcif 4479  𝒫 cpw 4554  ∪ cuni 4863   class class class wbr 5098  ran crn 5625  ℩cio 6446   defAt wdfat 47358  ''''cafv2 47450

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-ext 2708

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-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-dfat 47361  df-afv2 47451

This theorem is referenced by:  afv2eq1  47458  afv2eq2  47459  csbafv212g  47461