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Theorem revval 11719
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 2907 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5668 . . . . 5  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
32oveq2d 6036 . . . 4  |-  ( w  =  W  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  W
) ) )
4 id 20 . . . . 5  |-  ( w  =  W  ->  w  =  W )
52oveq1d 6035 . . . . . 6  |-  ( w  =  W  ->  (
( # `  w )  -  1 )  =  ( ( # `  W
)  -  1 ) )
65oveq1d 6035 . . . . 5  |-  ( w  =  W  ->  (
( ( # `  w
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  x ) )
74, 6fveq12d 5674 . . . 4  |-  ( w  =  W  ->  (
w `  ( (
( # `  w )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  x ) ) )
83, 7mpteq12dv 4228 . . 3  |-  ( w  =  W  ->  (
x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W ) )  |->  ( W `  ( ( ( # `  W
)  -  1 )  -  x ) ) ) )
9 df-reverse 11655 . . 3  |- reverse  =  ( w  e.  _V  |->  ( x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) ) )
10 ovex 6045 . . . 4  |-  ( 0..^ ( # `  W
) )  e.  _V
1110mptex 5905 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  e.  _V
128, 9, 11fvmpt 5745 . 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 set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2899    e. cmpt 4207   ` cfv 5394  (class class class)co 6020   0cc0 8923   1c1 8924    - cmin 9223  ..^cfzo 11065   #chash 11545  reversecreverse 11649
This theorem is referenced by:  revcl  11720  revlen  11721  revfv  11722  revco  11730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-reverse 11655
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