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Theorem symgextfve 18483
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 6669 . . 3 (𝑋 = 𝐾 → (𝐸𝑋) = (𝐸𝐾))
2 iftrue 4476 . . . . 5 (𝑥 = 𝐾 → if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)) = 𝐾)
3 symgext.e . . . . 5 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
42, 3fvmptg 6765 . . . 4 ((𝐾𝑁𝐾𝑁) → (𝐸𝐾) = 𝐾)
54anidms 567 . . 3 (𝐾𝑁 → (𝐸𝐾) = 𝐾)
61, 5sylan9eqr 2883 . 2 ((𝐾𝑁𝑋 = 𝐾) → (𝐸𝑋) = 𝐾)
76ex 413 1 (𝐾𝑁 → (𝑋 = 𝐾 → (𝐸𝑋) = 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  cdif 3937  ifcif 4470  {csn 4564  cmpt 5143  cfv 6354  Basecbs 16478  SymGrpcsymg 18440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362
This theorem is referenced by:  symgextf1lem  18484  symgextfo  18486
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