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 ) ) ) |