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

Theorem symgextfve 19209
Description: The function value of the extension of a permutation, fixing the additional element, for the additional element. (Contributed by AV, 6-Jan-2019.)
Hypotheses
Ref Expression
symgext.s 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
symgext.e 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
Assertion
Ref Expression
symgextfve (𝐾𝑁 → (𝑋 = 𝐾 → (𝐸𝑋) = 𝐾))
Distinct variable groups:   𝑥,𝐾   𝑥,𝑁   𝑥,𝑆   𝑥,𝑍   𝑥,𝑋
Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem symgextfve
StepHypRef Expression
1 fveq2 6846 . . 3 (𝑋 = 𝐾 → (𝐸𝑋) = (𝐸𝐾))
2 iftrue 4496 . . . . 5 (𝑥 = 𝐾 → if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)) = 𝐾)
3 symgext.e . . . . 5 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
42, 3fvmptg 6950 . . . 4 ((𝐾𝑁𝐾𝑁) → (𝐸𝐾) = 𝐾)
54anidms 568 . . 3 (𝐾𝑁 → (𝐸𝐾) = 𝐾)
61, 5sylan9eqr 2795 . 2 ((𝐾𝑁𝑋 = 𝐾) → (𝐸𝑋) = 𝐾)
76ex 414 1 (𝐾𝑁 → (𝑋 = 𝐾 → (𝐸𝑋) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  cdif 3911  ifcif 4490  {csn 4590  cmpt 5192  cfv 6500  Basecbs 17091  SymGrpcsymg 19156
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 5260  ax-nul 5267  ax-pr 5388
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-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508
This theorem is referenced by:  symgextf1lem  19210  symgextfo  19212
  Copyright terms: Public domain W3C validator