Theorem afv2eq12d 47130

Description: Equality deduction for function value, analogous to fveq12d 6927. (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 47041	. . 3 ⊢ (𝜑 → (𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵))
4		eqidd 2741	. . . . 5 ⊢ (𝜑 → 𝑥 = 𝑥)
5	2, 1, 4	breq123d 5180	. . . 4 ⊢ (𝜑 → (𝐴𝐹𝑥 ↔ 𝐵𝐺𝑥))
6	5	iotabidv 6557	. . 3 ⊢ (𝜑 → (℩𝑥𝐴𝐹𝑥) = (℩𝑥𝐵𝐺𝑥))
7	1	rneqd 5963	. . . . 5 ⊢ (𝜑 → ran 𝐹 = ran 𝐺)
8	7	unieqd 4944	. . . 4 ⊢ (𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺)
9	8	pweqd 4639	. . 3 ⊢ (𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺)
10	3, 6, 9	ifbieq12d 4576	. 2 ⊢ (𝜑 → if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺))
11		df-afv2 47124	. 2 ⊢ (𝐹''''𝐴) = if(𝐹 defAt 𝐴, (℩𝑥𝐴𝐹𝑥), 𝒫 ∪ ran 𝐹)
12		df-afv2 47124	. 2 ⊢ (𝐺''''𝐵) = if(𝐺 defAt 𝐵, (℩𝑥𝐵𝐺𝑥), 𝒫 ∪ ran 𝐺)
13	10, 11, 12	3eqtr4g 2805	1 ⊢ (𝜑 → (𝐹''''𝐴) = (𝐺''''𝐵))

Syntax hints:   → wi 4   = wceq 1537  ifcif 4548  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166  ran crn 5701  ℩cio 6523   defAt wdfat 47031  ''''cafv2 47123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-iota 6525  df-fun 6575  df-dfat 47034  df-afv2 47124

This theorem is referenced by:  afv2eq1  47131  afv2eq2  47132  csbafv212g  47134