Theorem derangval 31149

Description: Define the derangement function, which counts the number of bijections from a set to itself such that no element is mapped to itself. (Contributed by Mario Carneiro, 19-Jan-2015.)

Hypothesis

Ref	Expression
derang.d	|- D = ( x e. Fin |-> ( # ` { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } ) )

Assertion

Ref	Expression
derangval	|- ( A e. Fin -> ( D ` A ) = ( # ` { f | ( f : A -1-1-onto-> A /\ A. y e. A ( f ` y ) =/= y ) } ) )

Distinct variable group:   x, f, y, A

Allowed substitution hints:   D( x, y, f)

Proof of Theorem derangval

Step	Hyp	Ref	Expression
1		f1oeq2 6128	. . . . . 6 |- ( x = A -> ( f : x -1-1-onto-> x <-> f : A -1-1-onto-> x ) )
2		f1oeq3 6129	. . . . . 6 |- ( x = A -> ( f : A -1-1-onto-> x <-> f : A -1-1-onto-> A ) )
3	1, 2	bitrd 268	. . . . 5 |- ( x = A -> ( f : x -1-1-onto-> x <-> f : A -1-1-onto-> A ) )
4		raleq 3138	. . . . 5 |- ( x = A -> ( A. y e. x ( f ` y ) =/= y <-> A. y e. A ( f ` y ) =/= y ) )
5	3, 4	anbi12d 747	. . . 4 |- ( x = A -> ( ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) <-> ( f : A -1-1-onto-> A /\ A. y e. A ( f ` y ) =/= y ) ) )
6	5	abbidv 2741	. . 3 |- ( x = A -> { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } = { f | ( f : A -1-1-onto-> A /\ A. y e. A ( f ` y ) =/= y ) } )
7	6	fveq2d 6195	. 2 |- ( x = A -> ( # ` { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } ) = ( # ` { f | ( f : A -1-1-onto-> A /\ A. y e. A ( f ` y ) =/= y ) } ) )
8		derang.d	. 2 |- D = ( x e. Fin |-> ( # ` { f | ( f : x -1-1-onto-> x /\ A. y e. x ( f ` y ) =/= y ) } ) )
9		fvex 6201	. 2 |- ( # ` { f | ( f : A -1-1-onto-> A /\ A. y e. A ( f ` y ) =/= y ) } ) e. _V
10	7, 8, 9	fvmpt 6282	1 |- ( A e. Fin -> ( D ` A ) = ( # ` { f | ( f : A -1-1-onto-> A /\ A. y e. A ( f ` y ) =/= y ) } ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   =/= wne 2794   A.wral 2912   |-> cmpt 4729   -1-1-onto->wf1o 5887   ` cfv 5888   Fincfn 7955   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  derang0  31151  derangsn  31152  derangenlem  31153  subfaclefac  31158  subfacp1lem3  31164  subfacp1lem5  31166  subfacp1lem6  31167