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Metamath Proof Explorer

Theorem symgextsymg

Description: The extension of a permutation is an element of the extended symmetric group. (Contributed by AV, 9-Mar-2019)

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

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 1 2 symgextf1o 
 |-  ( ( K e. N /\ Z e. S ) -> E : N -1-1-onto-> N )
4 3 3adant1 
 |-  ( ( N e. V /\ K e. N /\ Z e. S ) -> E : N -1-1-onto-> N )
5 eqid 
 |-  ( SymGrp ` N ) = ( SymGrp ` N )
6 eqid 
 |-  ( Base ` ( SymGrp ` N ) ) = ( Base ` ( SymGrp ` N ) )
7 5 6 elsymgbas 
 |-  ( N e. V -> ( E e. ( Base ` ( SymGrp ` N ) ) <-> E : N -1-1-onto-> N ) )
8 7 3ad2ant1 
 |-  ( ( N e. V /\ K e. N /\ Z e. S ) -> ( E e. ( Base ` ( SymGrp ` N ) ) <-> E : N -1-1-onto-> N ) )
9 4 8 mpbird 
 |-  ( ( N e. V /\ K e. N /\ Z e. S ) -> E e. ( Base ` ( SymGrp ` N ) ) )