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Theorem nvocnvb 41674
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 7223 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
2 fveq1 6839 . . . . . 6 (𝐹 = 𝐹 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑥)))
32ad2antlr 725 . . . . 5 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑥)))
4 f1ocnvfv1 7219 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
51, 4sylan 580 . . . . 5 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
63, 5eqtr3d 2778 . . . 4 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
76ralrimiva 3142 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
81, 7jca 512 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥))
9 f1of 6782 . . 3 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
10 ffn 6666 . . . . 5 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
1110adantr 481 . . . 4 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 Fn 𝐴)
12 nvocnv 7224 . . . 4 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
1311, 12jca 512 . . 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 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 396   = wceq 1541  wcel 2106  wral 3063  ccnv 5631   Fn wfn 6489  wf 6490  1-1-ontowf1o 6493  cfv 6494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5530  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6446  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502
This theorem is referenced by: (None)
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