Theorem csbwrdg 13334

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.

Step	Hyp	Ref	Expression
1		df-word 13299	. . 3 |- Word x = { w | E. l e. NN0 w : ( 0..^ l ) --> x }
2	1	csbeq2i 3993	. 2 |- [_ S / x ]_Word x = [_ S / x ]_ { w | E. l e. NN0 w : ( 0..^ l ) --> x }
3		sbcrex 3514	. . . . 5 |- ( [. S / x ]. E. l e. NN0 w : ( 0..^ l ) --> x <-> E. l e. NN0 [. S / x ]. w : ( 0..^ l ) --> x )
4		sbcfg 6043	. . . . . . 7 |- ( S e. V -> ( [. S / x ]. w : ( 0..^ l ) --> x <-> [_ S / x ]_ w : [_ S / x ]_ ( 0..^ l ) --> [_ S / x ]_ x ) )
5		csbconstg 3546	. . . . . . . 8 |- ( S e. V -> [_ S / x ]_ w = w )
6		csbconstg 3546	. . . . . . . 8 |- ( S e. V -> [_ S / x ]_ ( 0..^ l ) = ( 0..^ l ) )
7		csbvarg 4003	. . . . . . . 8 |- ( S e. V -> [_ S / x ]_ x = S )
8	5, 6, 7	feq123d 6034	. . . . . . 7 |- ( S e. V -> ( [_ S / x ]_ w : [_ S / x ]_ ( 0..^ l ) --> [_ S / x ]_ x <-> w : ( 0..^ l ) --> S ) )
9	4, 8	bitrd 268	. . . . . 6 |- ( S e. V -> ( [. S / x ]. w : ( 0..^ l ) --> x <-> w : ( 0..^ l ) --> S ) )
10	9	rexbidv 3052	. . . . 5 |- ( S e. V -> ( E. l e. NN0 [. S / x ]. w : ( 0..^ l ) --> x <-> E. l e. NN0 w : ( 0..^ l ) --> S ) )
11	3, 10	syl5bb 272	. . . 4 |- ( S e. V -> ( [. S / x ]. E. l e. NN0 w : ( 0..^ l ) --> x <-> E. l e. NN0 w : ( 0..^ l ) --> S ) )
12	11	abbidv 2741	. . 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 4008	. . 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 13299	. . 3 |- Word S = { w | E. l e. NN0 w : ( 0..^ l ) --> S }
15	12, 13, 14	3eqtr4g 2681	. 2 |- ( S e. V -> [_ S / x ]_ { w | E. l e. NN0 w : ( 0..^ l ) --> x } = Word S )
16	2, 15	syl5eq 2668	1 |- ( S e. V -> [_ S / x ]_Word x = Word S )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   [.wsbc 3435   [_csb 3533   -->wf 5884  (class class class)co 6650   0cc0 9936   NN0cn0 11292  ..^cfzo 12465  Word cword 13291

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-fun 5890  df-fn 5891  df-f 5892  df-word 13299

This theorem is referenced by:  elovmpt2wrd  13347