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Theorem nvocnv 6494
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 795 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑦 = (𝐹𝑧))
2 simpll 789 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝐹:𝐴𝐴)
3 simprl 793 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑧𝐴)
42, 3ffvelrnd 6318 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑧) ∈ 𝐴)
51, 4eqeltrd 2698 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑦𝐴)
61fveq2d 6154 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑦) = (𝐹‘(𝐹𝑧)))
7 simplr 791 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
8 fveq2 6150 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝐹𝑥) = (𝐹𝑧))
98fveq2d 6154 . . . . . . . . 9 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
10 id 22 . . . . . . . . 9 (𝑥 = 𝑧𝑥 = 𝑧)
119, 10eqeq12d 2636 . . . . . . . 8 (𝑥 = 𝑧 → ((𝐹‘(𝐹𝑥)) = 𝑥 ↔ (𝐹‘(𝐹𝑧)) = 𝑧))
1211rspcv 3291 . . . . . . 7 (𝑧𝐴 → (∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥 → (𝐹‘(𝐹𝑧)) = 𝑧))
133, 7, 12sylc 65 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹‘(𝐹𝑧)) = 𝑧)
146, 13eqtr2d 2656 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑧 = (𝐹𝑦))
155, 14jca 554 . . . 4 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝑦𝐴𝑧 = (𝐹𝑦)))
16 simprr 795 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑧 = (𝐹𝑦))
17 simpll 789 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝐹:𝐴𝐴)
18 simprl 793 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑦𝐴)
1917, 18ffvelrnd 6318 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑦) ∈ 𝐴)
2016, 19eqeltrd 2698 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑧𝐴)
2116fveq2d 6154 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑧) = (𝐹‘(𝐹𝑦)))
22 simplr 791 . . . . . . 7 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥)
23 fveq2 6150 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝐹𝑥) = (𝐹𝑦))
2423fveq2d 6154 . . . . . . . . 9 (𝑥 = 𝑦 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑦)))
25 id 22 . . . . . . . . 9 (𝑥 = 𝑦𝑥 = 𝑦)
2624, 25eqeq12d 2636 . . . . . . . 8 (𝑥 = 𝑦 → ((𝐹‘(𝐹𝑥)) = 𝑥 ↔ (𝐹‘(𝐹𝑦)) = 𝑦))
2726rspcv 3291 . . . . . . 7 (𝑦𝐴 → (∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥 → (𝐹‘(𝐹𝑦)) = 𝑦))
2818, 22, 27sylc 65 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹‘(𝐹𝑦)) = 𝑦)
2921, 28eqtr2d 2656 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑦 = (𝐹𝑧))
3020, 29jca 554 . . . 4 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝑧𝐴𝑦 = (𝐹𝑧)))
3115, 30impbida 876 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → ((𝑧𝐴𝑦 = (𝐹𝑧)) ↔ (𝑦𝐴𝑧 = (𝐹𝑦))))
3231mptcnv 5495 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑦𝐴 ↦ (𝐹𝑦)))
33 ffn 6004 . . . 4 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
34 dffn5 6200 . . . . . 6 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3534biimpi 206 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3635adantr 481 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3733, 36sylan 488 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3837cnveqd 5260 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
39 dffn5 6200 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
4039biimpi 206 . . . 4 (𝐹 Fn 𝐴𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
4140adantr 481 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
4233, 41sylan 488 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
4332, 38, 423eqtr4d 2665 1 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2907  cmpt 4675  ccnv 5075   Fn wfn 5844  wf 5845  cfv 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857
This theorem is referenced by:  mirf1o  25471  lmif1o  25594
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