Theorem f1oeq2d

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

	Ref	Expression
Hypothesis	f1oeq2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	f1oeq2d	⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		f1oeq2d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		f1oeq2	⊢ ( 𝐴 = 𝐵 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐶 ) )