Theorem f1oeq3d

Description: Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020)

	Ref	Expression
Hypothesis	f1oeq3d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	f1oeq3d	⊢ ( 𝜑 → ( 𝐹 : 𝐶 –1-1-onto→ 𝐴 ↔ 𝐹 : 𝐶 –1-1-onto→ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		f1oeq3d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		f1oeq3	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐶 –1-1-onto→ 𝐴 ↔ 𝐹 : 𝐶 –1-1-onto→ 𝐵 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐹 : 𝐶 –1-1-onto→ 𝐴 ↔ 𝐹 : 𝐶 –1-1-onto→ 𝐵 ) )