Theorem symgextfve 17839

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 e. N |-> if ( x = K , K , ( Z ` x ) ) )

Assertion

Ref	Expression
symgextfve	|- ( K e. 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 6191	. . 3 |- ( X = K -> ( E ` X ) = ( E ` K ) )
2		iftrue 4092	. . . . 5 |- ( x = K -> if ( x = K , K , ( Z ` x ) ) = K )
3		symgext.e	. . . . 5 |- E = ( x e. N |-> if ( x = K , K , ( Z ` x ) ) )
4	2, 3	fvmptg 6280	. . . 4 |- ( ( K e. N /\ K e. N ) -> ( E ` K ) = K )
5	4	anidms 677	. . 3 |- ( K e. N -> ( E ` K ) = K )
6	1, 5	sylan9eqr 2678	. 2 |- ( ( K e. N /\ X = K ) -> ( E ` X ) = K )
7	6	ex 450	1 |- ( K e. N -> ( X = K -> ( E ` X ) = K ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   \ cdif 3571   ifcif 4086   {csn 4177   |-> cmpt 4729   ` cfv 5888   Basecbs 15857   SymGrpcsymg 17797

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  symgextf1lem  17840  symgextfo  17842