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Theorem symgextfv 18538
Description: The function value of the extension of a permutation, fixing the additional element, for elements in the original domain. (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
symgextfv ((𝐾𝑁𝑍𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑋) = (𝑍𝑋)))
Distinct variable groups:   𝑥,𝐾   𝑥,𝑁   𝑥,𝑆   𝑥,𝑍   𝑥,𝑋
Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem symgextfv
StepHypRef Expression
1 eldifi 4101 . . . 4 (𝑋 ∈ (𝑁 ∖ {𝐾}) → 𝑋𝑁)
2 fvexd 6678 . . . . 5 ((𝐾𝑁𝑍𝑆) → (𝑍𝑋) ∈ V)
3 ifexg 4512 . . . . 5 ((𝐾𝑁 ∧ (𝑍𝑋) ∈ V) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
42, 3syldan 593 . . . 4 ((𝐾𝑁𝑍𝑆) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
5 eqeq1 2823 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = 𝐾𝑋 = 𝐾))
6 fveq2 6663 . . . . . 6 (𝑥 = 𝑋 → (𝑍𝑥) = (𝑍𝑋))
75, 6ifbieq2d 4490 . . . . 5 (𝑥 = 𝑋 → if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
8 symgext.e . . . . 5 𝐸 = (𝑥𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍𝑥)))
97, 8fvmptg 6759 . . . 4 ((𝑋𝑁 ∧ if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
101, 4, 9syl2anr 598 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
11 eldifsnneq 4715 . . . . 5 (𝑋 ∈ (𝑁 ∖ {𝐾}) → ¬ 𝑋 = 𝐾)
1211adantl 484 . . . 4 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ¬ 𝑋 = 𝐾)
1312iffalsed 4476 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) = (𝑍𝑋))
1410, 13eqtrd 2854 . 2 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = (𝑍𝑋))
1514ex 415 1 ((𝐾𝑁𝑍𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑋) = (𝑍𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398   = wceq 1530  wcel 2107  Vcvv 3493  cdif 3931  ifcif 4465  {csn 4559  cmpt 5137  cfv 6348  Basecbs 16475  SymGrpcsymg 18487
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 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356
This theorem is referenced by:  symgextf1lem  18540  symgextf1  18541  symgextfo  18542  symgextres  18545
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