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Metamath Proof Explorer

Theorem f1oeq1d

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

Ref Expression
Hypothesis f1oeq1d.1 ⊢ ( 𝜑 → 𝐹 = 𝐺 )
Assertion f1oeq1d ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐵 ) )

Proof

Step Hyp Ref Expression
1 f1oeq1d.1 ⊢ ( 𝜑 → 𝐹 = 𝐺 )
2 f1oeq1 ⊢ ( 𝐹 = 𝐺 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐵 ) )
3 1 2 syl ⊢ ( 𝜑 → ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ↔ 𝐺 : 𝐴 –1-1-onto→ 𝐵 ) )