Theorem f1oeq3dd 33321

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

Hypotheses

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

Assertion

Ref	Expression
f1oeq3dd	⊢ (𝜑 → 𝐹:𝐶⟶1-1onto𝐵)

Proof of Theorem f1oeq3dd

Step	Hyp	Ref	Expression
1		f1oeq3dd.1	. 2 ⊢ (𝜑 → 𝐹:𝐶⟶1-1onto𝐴)
2		f1oeq3dd.2	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
3	2	f1oeq3d 6851	. 2 ⊢ (𝜑 → (𝐹:𝐶⟶1-1onto𝐴 ↔ 𝐹:𝐶⟶1-1onto𝐵))
4	1, 3	mpbid 235	1 ⊢ (𝜑 → 𝐹:𝐶⟶1-1onto𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ⟶1-1ontowf1o 6565

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ss 3901  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573

This theorem is used by:  fcobijfs2  33410