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Theorem symgextfve 17662
 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 6103 . . 3 (𝑋 = 𝐾 → (𝐸𝑋) = (𝐸𝐾))
2 iftrue 4042 . . . . 5 (𝑥 = 𝐾 → if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)) = 𝐾)
3 symgext.e . . . . 5 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
42, 3fvmptg 6189 . . . 4 ((𝐾𝑁𝐾𝑁) → (𝐸𝐾) = 𝐾)
54anidms 675 . . 3 (𝐾𝑁 → (𝐸𝐾) = 𝐾)
61, 5sylan9eqr 2666 . 2 ((𝐾𝑁𝑋 = 𝐾) → (𝐸𝑋) = 𝐾)
76ex 449 1 (𝐾𝑁 → (𝑋 = 𝐾 → (𝐸𝑋) = 𝐾))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∖ cdif 3537  ifcif 4036  {csn 4125   ↦ cmpt 4643  ‘cfv 5804  Basecbs 15695  SymGrpcsymg 17620 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812 This theorem is referenced by:  symgextf1lem  17663  symgextfo  17665
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