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Theorem revval 12850
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 3096 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5881 . . . . 5  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
32oveq2d 6321 . . . 4  |-  ( w  =  W  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  W
) ) )
4 id 23 . . . . 5  |-  ( w  =  W  ->  w  =  W )
52oveq1d 6320 . . . . . 6  |-  ( w  =  W  ->  (
( # `  w )  -  1 )  =  ( ( # `  W
)  -  1 ) )
65oveq1d 6320 . . . . 5  |-  ( w  =  W  ->  (
( ( # `  w
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  x ) )
74, 6fveq12d 5887 . . . 4  |-  ( w  =  W  ->  (
w `  ( (
( # `  w )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  x ) ) )
83, 7mpteq12dv 4504 . . 3  |-  ( w  =  W  ->  (
x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W ) )  |->  ( W `  ( ( ( # `  W
)  -  1 )  -  x ) ) ) )
9 df-reverse 12657 . . 3  |- reverse  =  ( w  e.  _V  |->  ( x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) ) )
10 ovex 6333 . . . 4  |-  ( 0..^ ( # `  W
) )  e.  _V
1110mptex 6151 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  e.  _V
128, 9, 11fvmpt 5964 . 2  |-  ( W  e.  _V  ->  (reverse `  W )  =  ( x  e.  ( 0..^ ( # `  W
) )  |->  ( W `
 ( ( (
# `  W )  -  1 )  -  x ) ) ) )
131, 12syl 17 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 1437    e. wcel 1870   _Vcvv 3087    |-> cmpt 4484   ` cfv 5601  (class class class)co 6305   0cc0 9538   1c1 9539    - cmin 9859  ..^cfzo 11913   #chash 12512  reversecreverse 12649
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 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pr 4661
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-ov 6308  df-reverse 12657
This theorem is referenced by:  revcl  12851  revlen  12852  revfv  12853  repswrevw  12874  revco  12916
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