Theorem symgextfve 19638

Description: The function value of the extension of a permutation, fixing the additional element, for the additional element. (Contributed by AV, 6-Jan-2019.)

Hypotheses

Ref	Expression
symgext.s	⊢ 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
symgext.e	⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))

Assertion

Ref	Expression
symgextfve	⊢ (𝐾 ∈ 𝑁 → (𝑋 = 𝐾 → (𝐸‘𝑋) = 𝐾))

Distinct variable groups:   𝑥,𝐾   𝑥,𝑁   𝑥,𝑆   𝑥,𝑍   𝑥,𝑋

Allowed substitution hint:   𝐸(𝑥)

Proof of Theorem symgextfve

Step	Hyp	Ref	Expression
1		fveq2 6885	. . 3 ⊢ (𝑋 = 𝐾 → (𝐸‘𝑋) = (𝐸‘𝐾))
2		iftrue 4487	. . . . 5 ⊢ (𝑥 = 𝐾 → if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)) = 𝐾)
3		symgext.e	. . . . 5 ⊢ 𝐸 = (𝑥 ∈ 𝑁 ↦ if(𝑥 = 𝐾, 𝐾, (𝑍‘𝑥)))
4	2, 3	fvmptg 6991	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁) → (𝐸‘𝐾) = 𝐾)
5	4	anidms 577	. . 3 ⊢ (𝐾 ∈ 𝑁 → (𝐸‘𝐾) = 𝐾)
6	1, 5	sylan9eqr 2817	. 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑋 = 𝐾) → (𝐸‘𝑋) = 𝐾)
7	6	ex 418	1 ⊢ (𝐾 ∈ 𝑁 → (𝑋 = 𝐾 → (𝐸‘𝑋) = 𝐾))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144   ∖ cdif 4045  ifcif 4481  {csn 4584   ↦ cmpt 5185  ‘cfv 6538  Basecbs 17392  SymGrpcsymg 19588

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546

This theorem is used by:  symgextf1lem  19639  symgextfo  19641