Theorem fneq12

Description: Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017)

	Ref	Expression
Assertion	fneq12	⊢ ( ( 𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ) → ( 𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ) → 𝐹 = 𝐺 )
2		simpr	⊢ ( ( 𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 )
3	1 2	fneq12d	⊢ ( ( 𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ) → ( 𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐵 ) )