Theorem f1oeq3

Description: Equality theorem for one-to-one onto functions. (Contributed by NM, 10-Feb-1997)

	Ref	Expression
Assertion	f1oeq3	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –1-1-onto→ 𝐴 ↔ 𝐹 : 𝐶 –1-1-onto→ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		f1eq3	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –1-1→ 𝐴 ↔ 𝐹 : 𝐶 –1-1→ 𝐵 ) )
2		foeq3	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –onto→ 𝐴 ↔ 𝐹 : 𝐶 –onto→ 𝐵 ) )
3	1 2	anbi12d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐹 : 𝐶 –1-1→ 𝐴 ∧ 𝐹 : 𝐶 –onto→ 𝐴 ) ↔ ( 𝐹 : 𝐶 –1-1→ 𝐵 ∧ 𝐹 : 𝐶 –onto→ 𝐵 ) ) )
4		df-f1o	⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝐴 ↔ ( 𝐹 : 𝐶 –1-1→ 𝐴 ∧ 𝐹 : 𝐶 –onto→ 𝐴 ) )
5		df-f1o	⊢ ( 𝐹 : 𝐶 –1-1-onto→ 𝐵 ↔ ( 𝐹 : 𝐶 –1-1→ 𝐵 ∧ 𝐹 : 𝐶 –onto→ 𝐵 ) )
6	3 4 5	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –1-1-onto→ 𝐴 ↔ 𝐹 : 𝐶 –1-1-onto→ 𝐵 ) )