Theorem foeq2

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

	Ref	Expression
Assertion	foeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –onto→ 𝐶 ↔ 𝐹 : 𝐵 –onto→ 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-fo	⊢ ( 𝐹 : 𝐴 –onto→ 𝐶 ↔ ( 𝐹 Fn 𝐴 ∧ ran 𝐹 = 𝐶 ) )
4		df-fo	⊢ ( 𝐹 : 𝐵 –onto→ 𝐶 ↔ ( 𝐹 Fn 𝐵 ∧ ran 𝐹 = 𝐶 ) )
5	2 3 4	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –onto→ 𝐶 ↔ 𝐹 : 𝐵 –onto→ 𝐶 ) )