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Theorem nvocnv 7026
 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 772 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑦 = (𝐹𝑧))
2 simpll 766 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝐹:𝐴𝐴)
3 simprl 770 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑧𝐴)
42, 3ffvelrnd 6839 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑧) ∈ 𝐴)
51, 4eqeltrd 2890 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑦𝐴)
61fveq2d 6659 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑦) = (𝐹‘(𝐹𝑧)))
7 2fveq3 6660 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
8 id 22 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
97, 8eqeq12d 2814 . . . . . . 7 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = 𝑥 ↔ (𝐹‘(𝐹𝑧)) = 𝑧))
10 simplr 768 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
119, 10, 3rspcdva 3574 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹‘(𝐹𝑧)) = 𝑧)
126, 11eqtr2d 2834 . . . . 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 772 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑧 = (𝐹𝑦))
15 simpll 766 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝐹:𝐴𝐴)
16 simprl 770 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑦𝐴)
1715, 16ffvelrnd 6839 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑦) ∈ 𝐴)
1814, 17eqeltrd 2890 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑧𝐴)
1914fveq2d 6659 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑧) = (𝐹‘(𝐹𝑦)))
20 2fveq3 6660 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑦)))
21 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
2220, 21eqeq12d 2814 . . . . . . 7 (𝑥 = 𝑦 → ((𝐹‘(𝐹𝑥)) = 𝑥 ↔ (𝐹‘(𝐹𝑦)) = 𝑦))
23 simplr 768 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
2422, 23, 16rspcdva 3574 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹‘(𝐹𝑦)) = 𝑦)
2519, 24eqtr2d 2834 . . . . 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 800 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → ((𝑧𝐴𝑦 = (𝐹𝑧)) ↔ (𝑦𝐴𝑧 = (𝐹𝑦))))
2827mptcnv 5969 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑦𝐴 ↦ (𝐹𝑦)))
29 ffn 6495 . . . 4 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
30 dffn5 6709 . . . . . 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 5714 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
35 dffn5 6709 . . . . 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 2843 1 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ↦ cmpt 5114  ◡ccnv 5522   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-sbc 3723  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-fv 6340 This theorem is referenced by:  mirf1o  26507  lmif1o  26633
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