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Theorem nvocnvb 41768
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 7231 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → 𝐹:𝐴1-1-onto𝐴)
2 fveq1 6846 . . . . . 6 (𝐹 = 𝐹 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑥)))
32ad2antlr 726 . . . . 5 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑥)))
4 f1ocnvfv1 7227 . . . . . 6 ((𝐹:𝐴1-1-onto𝐴𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
51, 4sylan 581 . . . . 5 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
63, 5eqtr3d 2779 . . . 4 (((𝐹 Fn 𝐴𝐹 = 𝐹) ∧ 𝑥𝐴) → (𝐹‘(𝐹𝑥)) = 𝑥)
76ralrimiva 3144 . . 3 ((𝐹 Fn 𝐴𝐹 = 𝐹) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
81, 7jca 513 . 2 ((𝐹 Fn 𝐴𝐹 = 𝐹) → (𝐹:𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥))
9 f1of 6789 . . 3 (𝐹:𝐴1-1-onto𝐴𝐹:𝐴𝐴)
10 ffn 6673 . . . . 5 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
1110adantr 482 . . . 4 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 Fn 𝐴)
12 nvocnv 7232 . . . 4 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
1311, 12jca 513 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝐹 Fn 𝐴𝐹 = 𝐹))
149, 13sylan 581 . 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 397   = wceq 1542  wcel 2107  wral 3065  ccnv 5637   Fn wfn 6496  wf 6497  1-1-ontowf1o 6500  cfv 6501
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509
This theorem is referenced by: (None)
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