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Theorem revval 12697
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 3122 . 2  |-  ( W  e.  V  ->  W  e.  _V )
2 fveq2 5866 . . . . 5  |-  ( w  =  W  ->  ( # `
 w )  =  ( # `  W
) )
32oveq2d 6300 . . . 4  |-  ( w  =  W  ->  (
0..^ ( # `  w
) )  =  ( 0..^ ( # `  W
) ) )
4 id 22 . . . . 5  |-  ( w  =  W  ->  w  =  W )
52oveq1d 6299 . . . . . 6  |-  ( w  =  W  ->  (
( # `  w )  -  1 )  =  ( ( # `  W
)  -  1 ) )
65oveq1d 6299 . . . . 5  |-  ( w  =  W  ->  (
( ( # `  w
)  -  1 )  -  x )  =  ( ( ( # `  W )  -  1 )  -  x ) )
74, 6fveq12d 5872 . . . 4  |-  ( w  =  W  ->  (
w `  ( (
( # `  w )  -  1 )  -  x ) )  =  ( W `  (
( ( # `  W
)  -  1 )  -  x ) ) )
83, 7mpteq12dv 4525 . . 3  |-  ( w  =  W  ->  (
x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) )  =  ( x  e.  ( 0..^ ( # `  W ) )  |->  ( W `  ( ( ( # `  W
)  -  1 )  -  x ) ) ) )
9 df-reverse 12514 . . 3  |- reverse  =  ( w  e.  _V  |->  ( x  e.  ( 0..^ ( # `  w
) )  |->  ( w `
 ( ( (
# `  w )  -  1 )  -  x ) ) ) )
10 ovex 6309 . . . 4  |-  ( 0..^ ( # `  W
) )  e.  _V
1110mptex 6131 . . 3  |-  ( x  e.  ( 0..^ (
# `  W )
)  |->  ( W `  ( ( ( # `  W )  -  1 )  -  x ) ) )  e.  _V
128, 9, 11fvmpt 5950 . 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 1379    e. wcel 1767   _Vcvv 3113    |-> cmpt 4505   ` cfv 5588  (class class class)co 6284   0cc0 9492   1c1 9493    - cmin 9805  ..^cfzo 11792   #chash 12373  reversecreverse 12506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-reverse 12514
This theorem is referenced by:  revcl  12698  revlen  12699  revfv  12700  repswrevw  12721  revco  12763
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