Mathbox for Mario Carneiro < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  derangval Structured version   Unicode version

Theorem derangval 24855
 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
Assertion
Ref Expression
derangval
Distinct variable group:   ,,,
Allowed substitution hints:   (,,)

Proof of Theorem derangval
StepHypRef Expression
1 f1oeq2 5668 . . . . . 6
2 f1oeq3 5669 . . . . . 6
31, 2bitrd 246 . . . . 5
4 raleq 2906 . . . . 5
53, 4anbi12d 693 . . . 4
65abbidv 2552 . . 3
76fveq2d 5734 . 2
8 derang.d . 2
9 fvex 5744 . 2
107, 8, 9fvmpt 5808 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  cab 2424   wne 2601  wral 2707   cmpt 4268  wf1o 5455  cfv 5456  cfn 7111  chash 11620 This theorem is referenced by:  derang0  24857  derangsn  24858  derangenlem  24859  subfaclefac  24864  subfacp1lem3  24870  subfacp1lem5  24872  subfacp1lem6  24873 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464
 Copyright terms: Public domain W3C validator