Theorem symgextf1o

Description: The extension of a permutation, fixing the additional element, is a bijection. (Contributed by AV, 7-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	symgextf1o	|- ( ( K e. N /\ Z e. S ) -> E : N -1-1-onto-> 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	symgextf1	|- ( ( K e. N /\ Z e. S ) -> E : N -1-1-> N )
4	1 2	symgextfo	|- ( ( K e. N /\ Z e. S ) -> E : N -onto-> N )
5		df-f1o	|- ( E : N -1-1-onto-> N <-> ( E : N -1-1-> N /\ E : N -onto-> N ) )
6	3 4 5	sylanbrc	|- ( ( K e. N /\ Z e. S ) -> E : N -1-1-onto-> N )