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Theorem csbwrdg 12672
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 12640 . . 3  |- Word  x  =  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> x }
21csbeq2i 3807 . 2  |-  [_ S  /  x ]_Word  x  =  [_ S  /  x ]_ { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> x }
3 sbcrex 3372 . . . . 5  |-  ( [. S  /  x ]. E. l  e.  NN0  w : ( 0..^ l ) --> x  <->  E. l  e.  NN0  [. S  /  x ]. w : ( 0..^ l ) --> x )
4 sbcfg 5735 . . . . . . 7  |-  ( S  e.  V  ->  ( [. S  /  x ]. w : ( 0..^ l ) --> x  <->  [_ S  /  x ]_ w : [_ S  /  x ]_ (
0..^ l ) --> [_ S  /  x ]_ x ) )
5 csbconstg 3405 . . . . . . . 8  |-  ( S  e.  V  ->  [_ S  /  x ]_ w  =  w )
6 csbconstg 3405 . . . . . . . 8  |-  ( S  e.  V  ->  [_ S  /  x ]_ ( 0..^ l )  =  ( 0..^ l ) )
7 csbvarg 3817 . . . . . . . 8  |-  ( S  e.  V  ->  [_ S  /  x ]_ x  =  S )
85, 6, 7feq123d 5727 . . . . . . 7  |-  ( S  e.  V  ->  ( [_ S  /  x ]_ w : [_ S  /  x ]_ ( 0..^ l ) --> [_ S  /  x ]_ x  <->  w :
( 0..^ l ) --> S ) )
94, 8bitrd 256 . . . . . 6  |-  ( S  e.  V  ->  ( [. S  /  x ]. w : ( 0..^ l ) --> x  <->  w :
( 0..^ l ) --> S ) )
109rexbidv 2937 . . . . 5  |-  ( S  e.  V  ->  ( E. l  e.  NN0  [. S  /  x ]. w : ( 0..^ l ) --> x  <->  E. l  e.  NN0  w : ( 0..^ l ) --> S ) )
113, 10syl5bb 260 . . . 4  |-  ( S  e.  V  ->  ( [. S  /  x ]. E. l  e.  NN0  w : ( 0..^ l ) --> x  <->  E. l  e.  NN0  w : ( 0..^ l ) --> S ) )
1211abbidv 2556 . . 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 3822 . . 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 12640 . . 3  |- Word  S  =  { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> S }
1512, 13, 143eqtr4g 2486 . 2  |-  ( S  e.  V  ->  [_ S  /  x ]_ { w  |  E. l  e.  NN0  w : ( 0..^ l ) --> x }  = Word  S )
162, 15syl5eq 2473 1  |-  ( S  e.  V  ->  [_ S  /  x ]_Word  x  = Word  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1867   {cab 2405   E.wrex 2774   [.wsbc 3296   [_csb 3392   -->wf 5588  (class class class)co 6296   0cc0 9528   NN0cn0 10858  ..^cfzo 11902  Word cword 12632
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 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5594  df-fn 5595  df-f 5596  df-word 12640
This theorem is referenced by:  elovmpt2wrd  12685
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