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Theorem nvocnv 7092
Description: The converse of an involution is the function itself. (Contributed by Thierry Arnoux, 7-May-2019.)
Assertion
Ref Expression
nvocnv ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem nvocnv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprr 773 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑦 = (𝐹𝑧))
2 simpll 767 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝐹:𝐴𝐴)
3 simprl 771 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑧𝐴)
42, 3ffvelrnd 6905 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑧) ∈ 𝐴)
51, 4eqeltrd 2838 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑦𝐴)
61fveq2d 6721 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑦) = (𝐹‘(𝐹𝑧)))
7 2fveq3 6722 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
8 id 22 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
97, 8eqeq12d 2753 . . . . . . 7 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = 𝑥 ↔ (𝐹‘(𝐹𝑧)) = 𝑧))
10 simplr 769 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
119, 10, 3rspcdva 3539 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹‘(𝐹𝑧)) = 𝑧)
126, 11eqtr2d 2778 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑧 = (𝐹𝑦))
135, 12jca 515 . . . 4 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝑦𝐴𝑧 = (𝐹𝑦)))
14 simprr 773 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑧 = (𝐹𝑦))
15 simpll 767 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝐹:𝐴𝐴)
16 simprl 771 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑦𝐴)
1715, 16ffvelrnd 6905 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑦) ∈ 𝐴)
1814, 17eqeltrd 2838 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑧𝐴)
1914fveq2d 6721 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑧) = (𝐹‘(𝐹𝑦)))
20 2fveq3 6722 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑦)))
21 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
2220, 21eqeq12d 2753 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹‘(𝐹𝑥)) = 𝑥 ↔ (𝐹‘(𝐹𝑦)) = 𝑦))
23 simplr 769 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
2422, 23, 16rspcdva 3539 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹‘(𝐹𝑦)) = 𝑦)
2519, 24eqtr2d 2778 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑦 = (𝐹𝑧))
2618, 25jca 515 . . . 4 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝑧𝐴𝑦 = (𝐹𝑧)))
2713, 26impbida 801 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → ((𝑧𝐴𝑦 = (𝐹𝑧)) ↔ (𝑦𝐴𝑧 = (𝐹𝑦))))
2827mptcnv 6003 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑦𝐴 ↦ (𝐹𝑦)))
29 ffn 6545 . . . 4 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
30 dffn5 6771 . . . . . 6 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3130biimpi 219 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3231adantr 484 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3329, 32sylan 583 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3433cnveqd 5744 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
35 dffn5 6771 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3635biimpi 219 . . . 4 (𝐹 Fn 𝐴𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3736adantr 484 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3829, 37sylan 583 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3928, 34, 383eqtr4d 2787 1 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1543  wcel 2110  wral 3061  cmpt 5135  ccnv 5550   Fn wfn 6375  wf 6376  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fv 6388
This theorem is referenced by:  mirf1o  26760  lmif1o  26886
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