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Theorem revval 11784
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 2956 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5720 . . . . 5  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
32oveq2d 6089 . . . 4  |-  ( w  =  W  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  W
) ) )
4 id 20 . . . . 5  |-  ( w  =  W  ->  w  =  W )
52oveq1d 6088 . . . . . 6  |-  ( w  =  W  ->  (
( # `  w )  -  1 )  =  ( ( # `  W
)  -  1 ) )
65oveq1d 6088 . . . . 5  |-  ( w  =  W  ->  (
( ( # `  w
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  x ) )
74, 6fveq12d 5726 . . . 4  |-  ( w  =  W  ->  (
w `  ( (
( # `  w )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  x ) ) )
83, 7mpteq12dv 4279 . . 3  |-  ( w  =  W  ->  (
x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W ) )  |->  ( W `  ( ( ( # `  W
)  -  1 )  -  x ) ) ) )
9 df-reverse 11720 . . 3  |- reverse  =  ( w  e.  _V  |->  ( x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) ) )
10 ovex 6098 . . . 4  |-  ( 0..^ ( # `  W
) )  e.  _V
1110mptex 5958 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  e.  _V
128, 9, 11fvmpt 5798 . 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 1652    e. wcel 1725   _Vcvv 2948    e. cmpt 4258   ` cfv 5446  (class class class)co 6073   0cc0 8982   1c1 8983    - cmin 9283  ..^cfzo 11127   #chash 11610  reversecreverse 11714
This theorem is referenced by:  revcl  11785  revlen  11786  revfv  11787  revco  11795
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-reverse 11720
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