Theorem afv2eq12d

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

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

Proof

Step	Hyp	Ref	Expression
1		afv2eq12d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
2		afv2eq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3	1 2	dfateq12d	⊢ ( 𝜑 → ( 𝐹 defAt 𝐴 ↔ 𝐺 defAt 𝐵 ) )
4		eqidd	⊢ ( 𝜑 → 𝑥 = 𝑥 )
5	2 1 4	breq123d	⊢ ( 𝜑 → ( 𝐴 𝐹 𝑥 ↔ 𝐵 𝐺 𝑥 ) )
6	5	iotabidv	⊢ ( 𝜑 → ( ℩ 𝑥 𝐴 𝐹 𝑥 ) = ( ℩ 𝑥 𝐵 𝐺 𝑥 ) )
7	1	rneqd	⊢ ( 𝜑 → ran 𝐹 = ran 𝐺 )
8	7	unieqd	⊢ ( 𝜑 → ∪ ran 𝐹 = ∪ ran 𝐺 )
9	8	pweqd	⊢ ( 𝜑 → 𝒫 ∪ ran 𝐹 = 𝒫 ∪ ran 𝐺 )
10	3 6 9	ifbieq12d	⊢ ( 𝜑 → if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 ) = if ( 𝐺 defAt 𝐵 , ( ℩ 𝑥 𝐵 𝐺 𝑥 ) , 𝒫 ∪ ran 𝐺 ) )
11		df-afv2	⊢ ( 𝐹 '''' 𝐴 ) = if ( 𝐹 defAt 𝐴 , ( ℩ 𝑥 𝐴 𝐹 𝑥 ) , 𝒫 ∪ ran 𝐹 )
12		df-afv2	⊢ ( 𝐺 '''' 𝐵 ) = if ( 𝐺 defAt 𝐵 , ( ℩ 𝑥 𝐵 𝐺 𝑥 ) , 𝒫 ∪ ran 𝐺 )
13	10 11 12	3eqtr4g	⊢ ( 𝜑 → ( 𝐹 '''' 𝐴 ) = ( 𝐺 '''' 𝐵 ) )