Theorem fveq1

Description: Equality theorem for function value. (Contributed by NM, 29-Dec-1996)

	Ref	Expression
Assertion	fveq1	⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		breq	⊢ ( 𝐹 = 𝐺 → ( 𝐴 𝐹 𝑥 ↔ 𝐴 𝐺 𝑥 ) )
2	1	iotabidv	⊢ ( 𝐹 = 𝐺 → ( ℩ 𝑥 𝐴 𝐹 𝑥 ) = ( ℩ 𝑥 𝐴 𝐺 𝑥 ) )
3		df-fv	⊢ ( 𝐹 ‘ 𝐴 ) = ( ℩ 𝑥 𝐴 𝐹 𝑥 )
4		df-fv	⊢ ( 𝐺 ‘ 𝐴 ) = ( ℩ 𝑥 𝐴 𝐺 𝑥 )
5	2 3 4	3eqtr4g	⊢ ( 𝐹 = 𝐺 → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝐴 ) )