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Theorem derangval 29719
 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 5814 . . . . . 6
2 f1oeq3 5815 . . . . . 6
31, 2bitrd 256 . . . . 5
4 raleq 3023 . . . . 5
53, 4anbi12d 715 . . . 4
65abbidv 2556 . . 3
76fveq2d 5876 . 2
8 derang.d . 2
9 fvex 5882 . 2
107, 8, 9fvmpt 5955 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 370   wceq 1437   wcel 1867  cab 2405   wne 2616  wral 2773   cmpt 4475  wf1o 5591  cfv 5592  cfn 7568  chash 12501 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652 This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600 This theorem is referenced by:  derang0  29721  derangsn  29722  derangenlem  29723  subfaclefac  29728  subfacp1lem3  29734  subfacp1lem5  29736  subfacp1lem6  29737
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