Theorem feq2

Description: Equality theorem for functions. (Contributed by NM, 1-Aug-1994)

	Ref	Expression
Assertion	feq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐹 : 𝐵 ⟶ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		fneq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐵 ) )
2	1	anbi1d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶 ) ↔ ( 𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶 ) ) )
3		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐶 ) )
4		df-f	⊢ ( 𝐹 : 𝐵 ⟶ 𝐶 ↔ ( 𝐹 Fn 𝐵 ∧ ran 𝐹 ⊆ 𝐶 ) )
5	2 3 4	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐹 : 𝐵 ⟶ 𝐶 ) )