Theorem f1eq2

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

	Ref	Expression
Assertion	f1eq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1→ 𝐶 ↔ 𝐹 : 𝐵 –1-1→ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		feq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐹 : 𝐵 ⟶ 𝐶 ) )
2	1	anbi1d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐹 : 𝐴 ⟶ 𝐶 ∧ Fun ◡ 𝐹 ) ↔ ( 𝐹 : 𝐵 ⟶ 𝐶 ∧ Fun ◡ 𝐹 ) ) )
3		df-f1	⊢ ( 𝐹 : 𝐴 –1-1→ 𝐶 ↔ ( 𝐹 : 𝐴 ⟶ 𝐶 ∧ Fun ◡ 𝐹 ) )
4		df-f1	⊢ ( 𝐹 : 𝐵 –1-1→ 𝐶 ↔ ( 𝐹 : 𝐵 ⟶ 𝐶 ∧ Fun ◡ 𝐹 ) )
5	2 3 4	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1→ 𝐶 ↔ 𝐹 : 𝐵 –1-1→ 𝐶 ) )