Theorem symgextfve 19394

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 6840	. . 3 ⊢ (𝑋 = 𝐾 → (𝐸‘𝑋) = (𝐸‘𝐾))
2		iftrue 4472	. . . . 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 6945	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁) → (𝐸‘𝐾) = 𝐾)
5	4	anidms 566	. . 3 ⊢ (𝐾 ∈ 𝑁 → (𝐸‘𝐾) = 𝐾)
6	1, 5	sylan9eqr 2793	. 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑋 = 𝐾) → (𝐸‘𝑋) = 𝐾)
7	6	ex 412	1 ⊢ (𝐾 ∈ 𝑁 → (𝑋 = 𝐾 → (𝐸‘𝑋) = 𝐾))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∖ cdif 3886  ifcif 4466  {csn 4567   ↦ cmpt 5166  ‘cfv 6498  Basecbs 17179  SymGrpcsymg 19344

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506

This theorem is referenced by:  symgextf1lem  19395  symgextfo  19397