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

Theorem nvocnv 7301
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, 3ffvelcdmd 7105 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑧) ∈ 𝐴)
51, 4eqeltrd 2841 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → 𝑦𝐴)
61fveq2d 6910 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑧𝐴𝑦 = (𝐹𝑧))) → (𝐹𝑦) = (𝐹‘(𝐹𝑧)))
7 2fveq3 6911 . . . . . . . 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 3623 . . . . . 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 511 . . . 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, 16ffvelcdmd 7105 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑦) ∈ 𝐴)
1814, 17eqeltrd 2841 . . . . 5 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → 𝑧𝐴)
1914fveq2d 6910 . . . . . 6 (((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) ∧ (𝑦𝐴𝑧 = (𝐹𝑦))) → (𝐹𝑧) = (𝐹‘(𝐹𝑦)))
20 2fveq3 6911 . . . . . . . 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 3623 . . . . . 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 511 . . . 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 6159 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → (𝑧𝐴 ↦ (𝐹𝑧)) = (𝑦𝐴 ↦ (𝐹𝑦)))
29 ffn 6736 . . . 4 (𝐹:𝐴𝐴𝐹 Fn 𝐴)
30 dffn5 6967 . . . . . 6 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3130biimpi 216 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3231adantr 480 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3329, 32sylan 580 . . 3 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
3433cnveqd 5886 . 2 ((𝐹:𝐴𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑧𝐴 ↦ (𝐹𝑧)))
35 dffn5 6967 . . . . 5 (𝐹 Fn 𝐴𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3635biimpi 216 . . . 4 (𝐹 Fn 𝐴𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3736adantr 480 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹‘(𝐹𝑥)) = 𝑥) → 𝐹 = (𝑦𝐴 ↦ (𝐹𝑦)))
3829, 37sylan 580 . 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 395   = wceq 1540  wcel 2108  wral 3061  cmpt 5225  ccnv 5684   Fn wfn 6556  wf 6557  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  mirf1o  28677  lmif1o  28803  nvocnvb  43435
  Copyright terms: Public domain W3C validator