Theorem f1oeq2

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

	Ref	Expression
Assertion	f1oeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		f1eq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1→ 𝐶 ↔ 𝐹 : 𝐵 –1-1→ 𝐶 ) )
2		foeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –onto→ 𝐶 ↔ 𝐹 : 𝐵 –onto→ 𝐶 ) )
3	1 2	anbi12d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐹 : 𝐴 –1-1→ 𝐶 ∧ 𝐹 : 𝐴 –onto→ 𝐶 ) ↔ ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 : 𝐵 –onto→ 𝐶 ) ) )
4		df-f1o	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ ( 𝐹 : 𝐴 –1-1→ 𝐶 ∧ 𝐹 : 𝐴 –onto→ 𝐶 ) )
5		df-f1o	⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ ( 𝐹 : 𝐵 –1-1→ 𝐶 ∧ 𝐹 : 𝐵 –onto→ 𝐶 ) )
6	3 4 5	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )