Theorem foeq3

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

	Ref	Expression
Assertion	foeq3	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –onto→ 𝐴 ↔ 𝐹 : 𝐶 –onto→ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		eqeq2	⊢ ( 𝐴 = 𝐵 → ( ran 𝐹 = 𝐴 ↔ ran 𝐹 = 𝐵 ) )
2	1	anbi2d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴 ) ↔ ( 𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵 ) ) )
3		df-fo	⊢ ( 𝐹 : 𝐶 –onto→ 𝐴 ↔ ( 𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐴 ) )
4		df-fo	⊢ ( 𝐹 : 𝐶 –onto→ 𝐵 ↔ ( 𝐹 Fn 𝐶 ∧ ran 𝐹 = 𝐵 ) )
5	2 3 4	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –onto→ 𝐴 ↔ 𝐹 : 𝐶 –onto→ 𝐵 ) )