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Theorem symgextfve 17820
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 6178 . . 3 (𝑋 = 𝐾 → (𝐸𝑋) = (𝐸𝐾))
2 iftrue 4083 . . . . 5 (𝑥 = 𝐾 → if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)) = 𝐾)
3 symgext.e . . . . 5 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
42, 3fvmptg 6267 . . . 4 ((𝐾𝑁𝐾𝑁) → (𝐸𝐾) = 𝐾)
54anidms 676 . . 3 (𝐾𝑁 → (𝐸𝐾) = 𝐾)
61, 5sylan9eqr 2676 . 2 ((𝐾𝑁𝑋 = 𝐾) → (𝐸𝑋) = 𝐾)
76ex 450 1 (𝐾𝑁 → (𝑋 = 𝐾 → (𝐸𝑋) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  cdif 3564  ifcif 4077  {csn 4168  cmpt 4720  cfv 5876  Basecbs 15838  SymGrpcsymg 17778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884
This theorem is referenced by:  symgextf1lem  17821  symgextfo  17823
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