MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  symgextfv Structured version   Visualization version   GIF version

Theorem symgextfv 19367
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 4123 . . . 4 (𝑋 ∈ (𝑁 ∖ {𝐾}) → 𝑋𝑁)
2 fvexd 6907 . . . . 5 ((𝐾𝑁𝑍𝑆) → (𝑍𝑋) ∈ V)
3 ifexg 4574 . . . . 5 ((𝐾𝑁 ∧ (𝑍𝑋) ∈ V) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
42, 3syldan 590 . . . 4 ((𝐾𝑁𝑍𝑆) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
5 eqeq1 2732 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = 𝐾𝑋 = 𝐾))
6 fveq2 6892 . . . . . 6 (𝑥 = 𝑋 → (𝑍𝑥) = (𝑍𝑋))
75, 6ifbieq2d 4551 . . . . 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 596 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
11 eldifsnneq 4791 . . . . 5 (𝑋 ∈ (𝑁 ∖ {𝐾}) → ¬ 𝑋 = 𝐾)
1211adantl 481 . . . 4 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ¬ 𝑋 = 𝐾)
1312iffalsed 4536 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) = (𝑍𝑋))
1410, 13eqtrd 2768 . 2 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = (𝑍𝑋))
1514ex 412 1 ((𝐾𝑁𝑍𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑋) = (𝑍𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wcel 2099  Vcvv 3470  cdif 3942  ifcif 4525  {csn 4625  cmpt 5226  cfv 6543  Basecbs 17174  SymGrpcsymg 19315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-iota 6495  df-fun 6545  df-fv 6551
This theorem is referenced by:  symgextf1lem  19369  symgextf1  19370  symgextfo  19371  symgextres  19374
  Copyright terms: Public domain W3C validator