Metamath Proof Explorer


Theorem psgndmfi

Description: For a finite base set, the permutation sign is defined for all permutations. (Contributed by Thierry Arnoux, 22-Aug-2020)

Ref Expression
Hypotheses psgndmfi.s S = pmSgn D
psgndmfi.g G = Base SymGrp D
Assertion psgndmfi D Fin S Fn G

Proof

Step Hyp Ref Expression
1 psgndmfi.s S = pmSgn D
2 psgndmfi.g G = Base SymGrp D
3 eqid SymGrp D = SymGrp D
4 eqid p G | dom p I Fin = p G | dom p I Fin
5 3 2 4 1 psgnfn S Fn p G | dom p I Fin
6 3 2 sygbasnfpfi D Fin p G dom p I Fin
7 6 ralrimiva D Fin p G dom p I Fin
8 rabid2 G = p G | dom p I Fin p G dom p I Fin
9 7 8 sylibr D Fin G = p G | dom p I Fin
10 9 eqcomd D Fin p G | dom p I Fin = G
11 10 fneq2d D Fin S Fn p G | dom p I Fin S Fn G
12 5 11 mpbii D Fin S Fn G