Metamath Proof Explorer


Theorem evpmval

Description: Value of the set of even permutations, the alternating group. (Contributed by Thierry Arnoux, 1-Nov-2023)

Ref Expression
Hypothesis evpmval.1
|- A = ( pmEven ` D )
Assertion evpmval
|- ( D e. V -> A = ( `' ( pmSgn ` D ) " { 1 } ) )

Proof

Step Hyp Ref Expression
1 evpmval.1
 |-  A = ( pmEven ` D )
2 elex
 |-  ( D e. V -> D e. _V )
3 fveq2
 |-  ( d = D -> ( pmSgn ` d ) = ( pmSgn ` D ) )
4 3 cnveqd
 |-  ( d = D -> `' ( pmSgn ` d ) = `' ( pmSgn ` D ) )
5 4 imaeq1d
 |-  ( d = D -> ( `' ( pmSgn ` d ) " { 1 } ) = ( `' ( pmSgn ` D ) " { 1 } ) )
6 df-evpm
 |-  pmEven = ( d e. _V |-> ( `' ( pmSgn ` d ) " { 1 } ) )
7 fvex
 |-  ( pmSgn ` D ) e. _V
8 7 cnvex
 |-  `' ( pmSgn ` D ) e. _V
9 8 imaex
 |-  ( `' ( pmSgn ` D ) " { 1 } ) e. _V
10 5 6 9 fvmpt
 |-  ( D e. _V -> ( pmEven ` D ) = ( `' ( pmSgn ` D ) " { 1 } ) )
11 2 10 syl
 |-  ( D e. V -> ( pmEven ` D ) = ( `' ( pmSgn ` D ) " { 1 } ) )
12 1 11 syl5eq
 |-  ( D e. V -> A = ( `' ( pmSgn ` D ) " { 1 } ) )