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Theorem symgextfv 19377
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 4119 . . . 4 (𝑋 ∈ (𝑁 ∖ {𝐾}) → 𝑋𝑁)
2 fvexd 6907 . . . . 5 ((𝐾𝑁𝑍𝑆) → (𝑍𝑋) ∈ V)
3 ifexg 4573 . . . . 5 ((𝐾𝑁 ∧ (𝑍𝑋) ∈ V) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
42, 3syldan 589 . . . 4 ((𝐾𝑁𝑍𝑆) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
5 eqeq1 2729 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = 𝐾𝑋 = 𝐾))
6 fveq2 6892 . . . . . 6 (𝑥 = 𝑋 → (𝑍𝑥) = (𝑍𝑋))
75, 6ifbieq2d 4550 . . . . 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 6998 . . . 4 ((𝑋𝑁 ∧ if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
101, 4, 9syl2anr 595 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
11 eldifsnneq 4790 . . . . 5 (𝑋 ∈ (𝑁 ∖ {𝐾}) → ¬ 𝑋 = 𝐾)
1211adantl 480 . . . 4 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ¬ 𝑋 = 𝐾)
1312iffalsed 4535 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) = (𝑍𝑋))
1410, 13eqtrd 2765 . 2 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = (𝑍𝑋))
1514ex 411 1 ((𝐾𝑁𝑍𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑋) = (𝑍𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  Vcvv 3463  cdif 3936  ifcif 4524  {csn 4624  cmpt 5226  cfv 6543  Basecbs 17179  SymGrpcsymg 19325
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  symgextf1lem  19379  symgextf1  19380  symgextfo  19381  symgextres  19384
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