Theorem f1oeq23

Description: Equality theorem for one-to-one onto functions. (Contributed by FL, 14-Jul-2012)

	Ref	Expression
Assertion	f1oeq23	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		f1oeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )
2		f1oeq3	⊢ ( 𝐶 = 𝐷 → ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐷 ) )
3	1 2	sylan9bb	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐷 ) )