Theorem symgextfve 18539

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 6663	. . 3 ⊢ (𝑋 = 𝐾 → (𝐸‘𝑋) = (𝐸‘𝐾))
2		iftrue 4471	. . . . 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 6759	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁) → (𝐸‘𝐾) = 𝐾)
5	4	anidms 569	. . 3 ⊢ (𝐾 ∈ 𝑁 → (𝐸‘𝐾) = 𝐾)
6	1, 5	sylan9eqr 2876	. 2 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝑋 = 𝐾) → (𝐸‘𝑋) = 𝐾)
7	6	ex 415	1 ⊢ (𝐾 ∈ 𝑁 → (𝑋 = 𝐾 → (𝐸‘𝑋) = 𝐾))

Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108   ∖ cdif 3931  ifcif 4465  {csn 4559   ↦ cmpt 5137  ‘cfv 6348  Basecbs 16475  SymGrpcsymg 18487

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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356

This theorem is referenced by:  symgextf1lem  18540  symgextfo  18542