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Theorem nvocnvb 43412
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 7300 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
2 fveq1 6906 . . . . . 6 (𝐹 = 𝐹 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑥)))
32ad2antlr 727 . . . . 5 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑥)))
4 f1ocnvfv1 7296 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
51, 4sylan 580 . . . . 5 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
63, 5eqtr3d 2777 . . . 4 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
76ralrimiva 3144 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
81, 7jca 511 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥))
9 f1of 6849 . . 3 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
10 ffn 6737 . . . . 5 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
1110adantr 480 . . . 4 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 Fn 𝐴)
12 nvocnv 7301 . . . 4 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
1311, 12jca 511 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝐹 Fn 𝐴𝐹 = 𝐹))
149, 13sylan 580 . 2 ((𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝐹 Fn 𝐴𝐹 = 𝐹))
158, 14impbii 209 1 ((𝐹 Fn 𝐴𝐹 = 𝐹) ↔ (𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  ccnv 5688   Fn wfn 6558  wf 6559  1-1-ontowf1o 6562  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571
This theorem is referenced by: (None)
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