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Theorem csbwrdg 12537
Description: Class substitution for the symbols of a word. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
csbwrdg  |-  ( S  e.  V  ->  [_ S  /  x ]_Word  x  = Word  S )
Distinct variable groups:    x, S    x, V

Proof of Theorem csbwrdg
Dummy variables  l  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-word 12509 . . 3  |- Word  x  =  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> x }
21csbeq2i 3836 . 2  |-  [_ S  /  x ]_Word  x  =  [_ S  /  x ]_ { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> x }
3 sbcrex 3416 . . . . 5  |-  ( [. S  /  x ]. E. l  e.  NN0  w : ( 0..^ l ) --> x  <->  E. l  e.  NN0  [. S  /  x ]. w : ( 0..^ l ) --> x )
4 sbcfg 5729 . . . . . . 7  |-  ( S  e.  V  ->  ( [. S  /  x ]. w : ( 0..^ l ) --> x  <->  [_ S  /  x ]_ w : [_ S  /  x ]_ (
0..^ l ) --> [_ S  /  x ]_ x ) )
5 csbconstg 3448 . . . . . . . 8  |-  ( S  e.  V  ->  [_ S  /  x ]_ w  =  w )
6 csbconstg 3448 . . . . . . . 8  |-  ( S  e.  V  ->  [_ S  /  x ]_ ( 0..^ l )  =  ( 0..^ l ) )
7 csbvarg 3848 . . . . . . . 8  |-  ( S  e.  V  ->  [_ S  /  x ]_ x  =  S )
85, 6, 7feq123d 5721 . . . . . . 7  |-  ( S  e.  V  ->  ( [_ S  /  x ]_ w : [_ S  /  x ]_ ( 0..^ l ) --> [_ S  /  x ]_ x  <->  w :
( 0..^ l ) --> S ) )
94, 8bitrd 253 . . . . . 6  |-  ( S  e.  V  ->  ( [. S  /  x ]. w : ( 0..^ l ) --> x  <->  w :
( 0..^ l ) --> S ) )
109rexbidv 2973 . . . . 5  |-  ( S  e.  V  ->  ( E. l  e.  NN0  [. S  /  x ]. w : ( 0..^ l ) --> x  <->  E. l  e.  NN0  w : ( 0..^ l ) --> S ) )
113, 10syl5bb 257 . . . 4  |-  ( S  e.  V  ->  ( [. S  /  x ]. E. l  e.  NN0  w : ( 0..^ l ) --> x  <->  E. l  e.  NN0  w : ( 0..^ l ) --> S ) )
1211abbidv 2603 . . 3  |-  ( S  e.  V  ->  { w  |  [. S  /  x ]. E. l  e.  NN0  w : ( 0..^ l ) --> x }  =  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> S } )
13 csbab 3855 . . 3  |-  [_ S  /  x ]_ { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> x }  =  { w  |  [. S  /  x ]. E. l  e.  NN0  w : ( 0..^ l ) --> x }
14 df-word 12509 . . 3  |- Word  S  =  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> S }
1512, 13, 143eqtr4g 2533 . 2  |-  ( S  e.  V  ->  [_ S  /  x ]_ { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> x }  = Word  S )
162, 15syl5eq 2520 1  |-  ( S  e.  V  ->  [_ S  /  x ]_Word  x  = Word  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815   [.wsbc 3331   [_csb 3435   -->wf 5584  (class class class)co 6285   0cc0 9493   NN0cn0 10796  ..^cfzo 11793  Word cword 12501
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-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-fal 1385  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-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-br 4448  df-opab 4506  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-fun 5590  df-fn 5591  df-f 5592  df-word 12509
This theorem is referenced by:  elovmpt2wrd  12549
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