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Theorem symgextfv 19387
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 4063 . . . 4 (𝑋 ∈ (𝑁 ∖ {𝐾}) → 𝑋𝑁)
2 fvexd 6845 . . . . 5 ((𝐾𝑁𝑍𝑆) → (𝑍𝑋) ∈ V)
3 ifexg 4506 . . . . 5 ((𝐾𝑁 ∧ (𝑍𝑋) ∈ V) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
42, 3syldan 598 . . . 4 ((𝐾𝑁𝑍𝑆) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
5 eqeq1 2745 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = 𝐾𝑋 = 𝐾))
6 fveq2 6830 . . . . . 6 (𝑥 = 𝑋 → (𝑍𝑥) = (𝑍𝑋))
75, 6ifbieq2d 4483 . . . . 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 6936 . . . 4 ((𝑋𝑁 ∧ if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
101, 4, 9syl2anr 604 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
11 eldifsnneq 4726 . . . . 5 (𝑋 ∈ (𝑁 ∖ {𝐾}) → ¬ 𝑋 = 𝐾)
1211adantl 483 . . . 4 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ¬ 𝑋 = 𝐾)
1312iffalsed 4467 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) = (𝑍𝑋))
1410, 13eqtrd 2776 . 2 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = (𝑍𝑋))
1514ex 414 1 ((𝐾𝑁𝑍𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑋) = (𝑍𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397   = wceq 1548  wcel 2121  Vcvv 3433  cdif 3881  ifcif 4456  {csn 4557  cmpt 5155  cfv 6488  Basecbs 17174  SymGrpcsymg 19338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5220  ax-nul 5230  ax-pr 5364
This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-iota 6444  df-fun 6490  df-fv 6496
This theorem is referenced by:  symgextf1lem  19389  symgextf1  19390  symgextfo  19391  symgextres  19394
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