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Theorem nvocnvb 42883
Description: Equivalence to saying the converse of an involution is the function itself. (Contributed by RP, 13-Oct-2024.)
Assertion
Ref Expression
nvocnvb ((𝐹 Fn 𝐴𝐹 = 𝐹) ↔ (𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem nvocnvb
StepHypRef Expression
1 nvof1o 7295 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
2 fveq1 6901 . . . . . 6 (𝐹 = 𝐹 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑥)))
32ad2antlr 725 . . . . 5 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑥)))
4 f1ocnvfv1 7291 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
51, 4sylan 578 . . . . 5 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
63, 5eqtr3d 2770 . . . 4 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
76ralrimiva 3143 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
81, 7jca 510 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥))
9 f1of 6844 . . 3 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
10 ffn 6727 . . . . 5 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
1110adantr 479 . . . 4 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 Fn 𝐴)
12 nvocnv 7296 . . . 4 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
1311, 12jca 510 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝐹 Fn 𝐴𝐹 = 𝐹))
149, 13sylan 578 . 2 ((𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝐹 Fn 𝐴𝐹 = 𝐹))
158, 14impbii 208 1 ((𝐹 Fn 𝐴𝐹 = 𝐹) ↔ (𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3058  ccnv 5681   Fn wfn 6548  wf 6549  1-1-ontowf1o 6552  cfv 6553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561
This theorem is referenced by: (None)
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