Theorem afv2eq12d 48421

Description: Equality deduction for function value, analogous to fveq12d 6897. (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 48332	. . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵))
4		eqidd 2761	. . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥)
5	2, 1, 4	breq123d 5116	. . . 4 ⊢ (𝜑 → (𝐴𝐹𝑥 ↔ 𝐵𝐺𝑥))
6	5	iotabidv 6526	. . 3 ⊢ (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
7	1	rneqd 5921	. . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐺)
8	7	unieqd 4879	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺)
9	8	pweqd 4573	. . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺)
10	3, 6, 9	ifbieq12d 4510	. 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺))
11		df-afv2 48415	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
12		df-afv2 48415	. 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)
13	10, 11, 12	3eqtr4g 2820	1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ifcif 4481  𝒫 cpw 4556  ∪ cuni 4866   class class class wbr 5102  ran crn 5652  ℩cio 6495   defAt wdfat 48323  ''''cafv2 48414

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6497  df-fun 6545  df-dfat 48326  df-afv2 48415

This theorem is used by:  afv2eq1  48422  afv2eq2  48423  csbafv212g  48425