Theorem f1eq3

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

	Ref	Expression
Assertion	f1eq3	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –1-1→ 𝐴 ↔ 𝐹 : 𝐶 –1-1→ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		feq3	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 ⟶ 𝐴 ↔ 𝐹 : 𝐶 ⟶ 𝐵 ) )
2	1	anbi1d	⊢ ( 𝐴 = 𝐵 → ( ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ Fun ◡ 𝐹 ) ↔ ( 𝐹 : 𝐶 ⟶ 𝐵 ∧ Fun ◡ 𝐹 ) ) )
3		df-f1	⊢ ( 𝐹 : 𝐶 –1-1→ 𝐴 ↔ ( 𝐹 : 𝐶 ⟶ 𝐴 ∧ Fun ◡ 𝐹 ) )
4		df-f1	⊢ ( 𝐹 : 𝐶 –1-1→ 𝐵 ↔ ( 𝐹 : 𝐶 ⟶ 𝐵 ∧ Fun ◡ 𝐹 ) )
5	2 3 4	3bitr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –1-1→ 𝐴 ↔ 𝐹 : 𝐶 –1-1→ 𝐵 ) )