Metamath Proof Explorer


Theorem symgressbas

Description: The symmetric group on A characterized as structure restriction of the monoid of endofunctions on A to its base set. (Contributed by AV, 30-Mar-2024)

Ref Expression
Hypotheses symgbas.1 G = SymGrp A
symgbas.2 B = Base G
symgressbas.m No typesetting found for |- M = ( EndoFMnd ` A ) with typecode |-
Assertion symgressbas G = M 𝑠 B

Proof

Step Hyp Ref Expression
1 symgbas.1 G = SymGrp A
2 symgbas.2 B = Base G
3 symgressbas.m Could not format M = ( EndoFMnd ` A ) : No typesetting found for |- M = ( EndoFMnd ` A ) with typecode |-
4 eqid f | f : A 1-1 onto A = f | f : A 1-1 onto A
5 1 4 symgval Could not format G = ( ( EndoFMnd ` A ) |`s { f | f : A -1-1-onto-> A } ) : No typesetting found for |- G = ( ( EndoFMnd ` A ) |`s { f | f : A -1-1-onto-> A } ) with typecode |-
6 1 2 symgbas B = f | f : A 1-1 onto A
7 3 6 oveq12i Could not format ( M |`s B ) = ( ( EndoFMnd ` A ) |`s { f | f : A -1-1-onto-> A } ) : No typesetting found for |- ( M |`s B ) = ( ( EndoFMnd ` A ) |`s { f | f : A -1-1-onto-> A } ) with typecode |-
8 5 7 eqtr4i G = M 𝑠 B