Description: The subfactorial is defined as the number of derangements (see derangval ) of the set ( 1 ... N ) . (Contributed by Mario Carneiro, 21-Jan-2015)
Ref | Expression | ||
---|---|---|---|
Hypotheses | derang.d | |- D = ( x e. Fin |-> ( # ` { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } ) ) |
|
subfac.n | |- S = ( n e. NN0 |-> ( D ` ( 1 ... n ) ) ) |
||
Assertion | subfacval | |- ( N e. NN0 -> ( S ` N ) = ( D ` ( 1 ... N ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | derang.d | |- D = ( x e. Fin |-> ( # ` { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } ) ) |
|
2 | subfac.n | |- S = ( n e. NN0 |-> ( D ` ( 1 ... n ) ) ) |
|
3 | oveq2 | |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) ) |
|
4 | 3 | fveq2d | |- ( n = N -> ( D ` ( 1 ... n ) ) = ( D ` ( 1 ... N ) ) ) |
5 | fvex | |- ( D ` ( 1 ... N ) ) e. _V |
|
6 | 4 2 5 | fvmpt | |- ( N e. NN0 -> ( S ` N ) = ( D ` ( 1 ... N ) ) ) |