Theorem f1oeq3dd 32603

Description: Equality deduction for one-to-one onto functions. (Contributed by Thierry Arnoux, 10-Jan-2026.)

Hypotheses

Ref	Expression
f1oeq3dd.1	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
f1oeq3dd.2	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
f1oeq3dd	⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)

Proof of Theorem f1oeq3dd

Step	Hyp	Ref	Expression
1		f1oeq3dd.1	. 2 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴)
2		f1oeq3dd.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	f1oeq3d 6755	. 2 ⊢ (𝜑 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵))
4	1, 3	mpbid 232	1 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)

Syntax hints:   → wi 4   = wceq 1541  –1-1-onto→wf1o 6475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-cleq 2723  df-ss 3914  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483

This theorem is referenced by:  fcobijfs2  32697