Theorem symgextfv

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

	Ref	Expression
Hypotheses	symgext.s	|- S = ( Base ` ( SymGrp ` ( N \ { K } ) ) )
	symgext.e	|- E = ( x e. N |-> if ( x = K , K , ( Z ` x ) ) )
Assertion	symgextfv	|- ( ( K e. N /\ Z e. S ) -> ( X e. ( N \ { K } ) -> ( E ` X ) = ( Z ` X ) ) )

Proof

Step	Hyp	Ref	Expression
1		symgext.s	|- S = ( Base ` ( SymGrp ` ( N \ { K } ) ) )
2		symgext.e	|- E = ( x e. N |-> if ( x = K , K , ( Z ` x ) ) )
3		eldifi	|- ( X e. ( N \ { K } ) -> X e. N )
4		fvexd	|- ( ( K e. N /\ Z e. S ) -> ( Z ` X ) e. _V )
5		ifexg	|- ( ( K e. N /\ ( Z ` X ) e. _V ) -> if ( X = K , K , ( Z ` X ) ) e. _V )
6	4 5	syldan	|- ( ( K e. N /\ Z e. S ) -> if ( X = K , K , ( Z ` X ) ) e. _V )
7		eqeq1	|- ( x = X -> ( x = K <-> X = K ) )
8		fveq2	|- ( x = X -> ( Z ` x ) = ( Z ` X ) )
9	7 8	ifbieq2d	|- ( x = X -> if ( x = K , K , ( Z ` x ) ) = if ( X = K , K , ( Z ` X ) ) )
10	9 2	fvmptg	|- ( ( X e. N /\ if ( X = K , K , ( Z ` X ) ) e. _V ) -> ( E ` X ) = if ( X = K , K , ( Z ` X ) ) )
11	3 6 10	syl2anr	|- ( ( ( K e. N /\ Z e. S ) /\ X e. ( N \ { K } ) ) -> ( E ` X ) = if ( X = K , K , ( Z ` X ) ) )
12		eldifsnneq	|- ( X e. ( N \ { K } ) -> -. X = K )
13	12	adantl	|- ( ( ( K e. N /\ Z e. S ) /\ X e. ( N \ { K } ) ) -> -. X = K )
14	13	iffalsed	|- ( ( ( K e. N /\ Z e. S ) /\ X e. ( N \ { K } ) ) -> if ( X = K , K , ( Z ` X ) ) = ( Z ` X ) )
15	11 14	eqtrd	|- ( ( ( K e. N /\ Z e. S ) /\ X e. ( N \ { K } ) ) -> ( E ` X ) = ( Z ` X ) )
16	15	ex	|- ( ( K e. N /\ Z e. S ) -> ( X e. ( N \ { K } ) -> ( E ` X ) = ( Z ` X ) ) )