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Theorem symgextfv 18045
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 3883 . . . 4 (𝑋 ∈ (𝑁 ∖ {𝐾}) → 𝑋𝑁)
2 fvexd 6344 . . . . 5 ((𝐾𝑁𝑍𝑆) → (𝑍𝑋) ∈ V)
3 ifexg 4296 . . . . 5 ((𝐾𝑁 ∧ (𝑍𝑋) ∈ V) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
42, 3syldan 579 . . . 4 ((𝐾𝑁𝑍𝑆) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V)
5 eqeq1 2775 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = 𝐾𝑋 = 𝐾))
6 fveq2 6332 . . . . . 6 (𝑥 = 𝑋 → (𝑍𝑥) = (𝑍𝑋))
75, 6ifbieq2d 4250 . . . . 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 6422 . . . 4 ((𝑋𝑁 ∧ if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) ∈ V) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
101, 4, 9syl2anr 584 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)))
11 eldifsn 4453 . . . . . 6 (𝑋 ∈ (𝑁 ∖ {𝐾}) ↔ (𝑋𝑁𝑋𝐾))
12 df-ne 2944 . . . . . . . 8 (𝑋𝐾 ↔ ¬ 𝑋 = 𝐾)
1312biimpi 206 . . . . . . 7 (𝑋𝐾 → ¬ 𝑋 = 𝐾)
1413adantl 467 . . . . . 6 ((𝑋𝑁𝑋𝐾) → ¬ 𝑋 = 𝐾)
1511, 14sylbi 207 . . . . 5 (𝑋 ∈ (𝑁 ∖ {𝐾}) → ¬ 𝑋 = 𝐾)
1615adantl 467 . . . 4 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → ¬ 𝑋 = 𝐾)
1716iffalsed 4236 . . 3 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → if(𝑋 = 𝐾, 𝐾, (𝑍𝑋)) = (𝑍𝑋))
1810, 17eqtrd 2805 . 2 (((𝐾𝑁𝑍𝑆) ∧ 𝑋 ∈ (𝑁 ∖ {𝐾})) → (𝐸𝑋) = (𝑍𝑋))
1918ex 397 1 ((𝐾𝑁𝑍𝑆) → (𝑋 ∈ (𝑁 ∖ {𝐾}) → (𝐸𝑋) = (𝑍𝑋)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  cdif 3720  ifcif 4225  {csn 4316  cmpt 4863  cfv 6031  Basecbs 16064  SymGrpcsymg 18004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-iota 5994  df-fun 6033  df-fv 6039
This theorem is referenced by:  symgextf1lem  18047  symgextf1  18048  symgextfo  18049  symgextres  18052
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