Theorem feq1

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

	Ref	Expression
Assertion	feq1	⊢ ( 𝐹 = 𝐺 → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ 𝐺 : 𝐴 ⟶ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		fneq1	⊢ ( 𝐹 = 𝐺 → ( 𝐹 Fn 𝐴 ↔ 𝐺 Fn 𝐴 ) )
2		rneq	⊢ ( 𝐹 = 𝐺 → ran 𝐹 = ran 𝐺 )
3	2	sseq1d	⊢ ( 𝐹 = 𝐺 → ( ran 𝐹 ⊆ 𝐵 ↔ ran 𝐺 ⊆ 𝐵 ) )
4	1 3	anbi12d	⊢ ( 𝐹 = 𝐺 → ( ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) ↔ ( 𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵 ) ) )
5		df-f	⊢ ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵 ) )
6		df-f	⊢ ( 𝐺 : 𝐴 ⟶ 𝐵 ↔ ( 𝐺 Fn 𝐴 ∧ ran 𝐺 ⊆ 𝐵 ) )
7	4 5 6	3bitr4g	⊢ ( 𝐹 = 𝐺 → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ 𝐺 : 𝐴 ⟶ 𝐵 ) )