Theorem nvof1ob 38224

Description: A class equal to its converse is a function on a class if and only if it is a bijection from that class onto itself (the reverse implication holds without the hypothesis, see f1ofn 6853). (Contributed by BJ, 8-Nov-2026.) Can shorten nvof1o 7317 using biimpac 484.

Assertion

Ref	Expression
nvof1ob	⊢ (◡𝐹 = 𝐹 → (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶1-1onto𝐴))

Proof of Theorem nvof1ob

Step	Hyp	Ref	Expression
1		dff1o4 6861	. . 3 ⊢ (𝐹:𝐴⟶1-1onto𝐴 ↔ (𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐴))
2		fneq1 6658	. . . 4 ⊢ (◡𝐹 = 𝐹 → (◡𝐹 Fn 𝐴 ↔ 𝐹 Fn 𝐴))
3	2	anbi2d 642	. . 3 ⊢ (◡𝐹 = 𝐹 → ((𝐹 Fn 𝐴 ∧ ◡𝐹 Fn 𝐴) ↔ (𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐴)))
4	1, 3	bitrid 286	. 2 ⊢ (◡𝐹 = 𝐹 → (𝐹:𝐴⟶1-1onto𝐴 ↔ (𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐴)))
5		anidm 575	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐹 Fn 𝐴) ↔ 𝐹 Fn 𝐴)
6	4, 5	bitr2di 291	1 ⊢ (◡𝐹 = 𝐹 → (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶1-1onto𝐴))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ^◡ccnv 5655   Fn wfn 6559  ⟶1-1ontowf1o 6563

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-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571

This theorem is used by: (None)