Theorem symgextfve 19520

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 6888	. . 3 ⊢ (𝑋 = 𝐾 → (𝐸‘𝑋) = (𝐸‘𝐾))
2		iftrue 4498	. . . . 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 6994	. . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ 𝐾 ∈ 𝑁) → (𝐸‘𝐾) = 𝐾)
5	4	anidms 577	. . 3 ⊢ (𝐾 ∈ 𝑁 → (𝐸‘𝐾) = 𝐾)
6	1, 5	sylan9eqr 2823	. 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 1570   ∈ wcel 2146   ∖ cdif 3905  ifcif 4492  {csn 4594   ↦ cmpt 5197  ‘cfv 6543  Basecbs 17294  SymGrpcsymg 19470

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551

This theorem is used by:  symgextf1lem  19521  symgextfo  19523