Theorem afv2eq12d 47675

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 47586	. . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵))
4		eqidd 2738	. . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥)
5	2, 1, 4	breq123d 5100	. . . 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 4864	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺)
9	8	pweqd 4559	. . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺)
10	3, 6, 9	ifbieq12d 4496	. 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺))
11		df-afv2 47669	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
12		df-afv2 47669	. 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)
13	10, 11, 12	3eqtr4g 2797	1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Syntax hints:   → wi 4   = wceq 1542  ifcif 4467  𝒫 cpw 4542  ∪ cuni 4851   class class class wbr 5086  ran crn 5625  ℩cio 6446   defAt wdfat 47576  ''''cafv2 47668

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  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 47579  df-afv2 47669

This theorem is referenced by:  afv2eq1  47676  afv2eq2  47677  csbafv212g  47679