MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvocnv Structured version   Visualization version   GIF version

Theorem nvocnv 7228
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, 3ffvelcdmd 7037 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑧) ∈ 𝐴)
51, 4eqeltrd 2834 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑦𝐴)
61fveq2d 6847 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑦) = (𝐹‘(𝐹𝑧)))
7 2fveq3 6848 . . . . . . . 8 (𝑥 = 𝑧 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑧)))
8 id 22 . . . . . . . 8 (𝑥 = 𝑧𝑥 = 𝑧)
97, 8eqeq12d 2749 . . . . . . 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 3581 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹‘(𝐹𝑧)) = 𝑧)
126, 11eqtr2d 2774 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑧 = (𝐹𝑦))
135, 12jca 513 . . . 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, 16ffvelcdmd 7037 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑦) ∈ 𝐴)
1814, 17eqeltrd 2834 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑧𝐴)
1914fveq2d 6847 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑧) = (𝐹‘(𝐹𝑦)))
20 2fveq3 6848 . . . . . . . 8 (𝑥 = 𝑦 → (𝐹‘(𝐹𝑥)) = (𝐹‘(𝐹𝑦)))
21 id 22 . . . . . . . 8 (𝑥 = 𝑦𝑥 = 𝑦)
2220, 21eqeq12d 2749 . . . . . . 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 3581 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹‘(𝐹𝑦)) = 𝑦)
2519, 24eqtr2d 2774 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑦 = (𝐹𝑧))
2618, 25jca 513 . . . 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 6093 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑦𝐴 ↦ (𝐹𝑦)))
29 ffn 6669 . . . 4 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
30 dffn5 6902 . . . . . 6 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3130biimpi 215 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3231adantr 482 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3329, 32sylan 581 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3433cnveqd 5832 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
35 dffn5 6902 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3635biimpi 215 . . . 4 (𝐹 Fn 𝐴𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3736adantr 482 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3829, 37sylan 581 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3928, 34, 383eqtr4d 2783 1 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  cmpt 5189  ccnv 5633   Fn wfn 6492  wf 6493  cfv 6497
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 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505
This theorem is referenced by:  mirf1o  27653  lmif1o  27779  nvocnvb  41782
  Copyright terms: Public domain W3C validator