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Theorem symgextfve 18148
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 6409 . . 3 (𝑋 = 𝐾 → (𝐸𝑋) = (𝐸𝐾))
2 iftrue 4281 . . . . 5 (𝑥 = 𝐾 → if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)) = 𝐾)
3 symgext.e . . . . 5 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
42, 3fvmptg 6503 . . . 4 ((𝐾𝑁𝐾𝑁) → (𝐸𝐾) = 𝐾)
54anidms 563 . . 3 (𝐾𝑁 → (𝐸𝐾) = 𝐾)
61, 5sylan9eqr 2853 . 2 ((𝐾𝑁𝑋 = 𝐾) → (𝐸𝑋) = 𝐾)
76ex 402 1 (𝐾𝑁 → (𝑋 = 𝐾 → (𝐸𝑋) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  cdif 3764  ifcif 4275  {csn 4366  cmpt 4920  cfv 6099  Basecbs 16181  SymGrpcsymg 18106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107
This theorem is referenced by:  symgextf1lem  18149  symgextfo  18151
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