Theorem f1oeq2d 5469

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

Hypothesis

Ref	Expression
f1oeq2d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

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

Proof of Theorem f1oeq2d

Step	Hyp	Ref	Expression
1		f1oeq2d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		f1oeq2 5462	. 2 ⊢ (𝐴 = 𝐵 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐶 ↔ 𝐹:𝐵–1-1-onto→𝐶))

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1363  –1-1-onto→wf1o 5227

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-gen 1459  ax-4 1520  ax-17 1536  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-cleq 2180  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235

This theorem is referenced by:  prodmodclem3  11596  prodmodc  11599  fprodseq  11604