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Theorem revval 12399
Description: Value of the word reversing function. (Contributed by Stefan O'Rear, 26-Aug-2015.)
Assertion
Ref Expression
revval  |-  ( W  e.  V  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
Distinct variable group:    x, W
Allowed substitution hint:    V( x)

Proof of Theorem revval
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 elex 2980 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5690 . . . . 5  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
32oveq2d 6106 . . . 4  |-  ( w  =  W  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  W
) ) )
4 id 22 . . . . 5  |-  ( w  =  W  ->  w  =  W )
52oveq1d 6105 . . . . . 6  |-  ( w  =  W  ->  (
( # `  w )  -  1 )  =  ( ( # `  W
)  -  1 ) )
65oveq1d 6105 . . . . 5  |-  ( w  =  W  ->  (
( ( # `  w
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  x ) )
74, 6fveq12d 5696 . . . 4  |-  ( w  =  W  ->  (
w `  ( (
( # `  w )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  x ) ) )
83, 7mpteq12dv 4369 . . 3  |-  ( w  =  W  ->  (
x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W ) )  |->  ( W `  ( ( ( # `  W
)  -  1 )  -  x ) ) ) )
9 df-reverse 12234 . . 3  |- reverse  =  ( w  e.  _V  |->  ( x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) ) )
10 ovex 6115 . . . 4  |-  ( 0..^ ( # `  W
) )  e.  _V
1110mptex 5947 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  e.  _V
128, 9, 11fvmpt 5773 . 2  |-  ( W  e.  _V  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
131, 12syl 16 1  |-  ( W  e.  V  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2971    e. cmpt 4349   ` cfv 5417  (class class class)co 6090   0cc0 9281   1c1 9282    - cmin 9594  ..^cfzo 11547   #chash 12102  reversecreverse 12226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-reverse 12234
This theorem is referenced by:  revcl  12400  revlen  12401  revfv  12402  repswrevw  12423  revco  12461
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