Theorem f1oeq3dd 33271

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 6822	. 2 ⊢ (𝜑 → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:𝐶–1-1-onto→𝐵))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1569  –1-1-onto→wf1o 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-ss 3896  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546

This theorem is used by:  fcobijfs2  33361