Theorem eclclwwlksn1 26952

Description: An equivalence class according to .~. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.)

Hypotheses

Ref	Expression
erclwwlksn.w	|- W = ( N ClWWalksN G )
erclwwlksn.r	|- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) }

Assertion

Ref	Expression
eclclwwlksn1	|- ( B e. X -> ( B e. ( W /. .~ ) <-> E. x e. W B = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) )

Distinct variable groups:   t, W, u   n, N, u, t, x, y   n, W   x, .~ , y   x, W   x, G   x, X   x, B, y   y, N   y, W   y, X

Allowed substitution hints:   B( u, t, n)   .~ ( u, t, n)   G( y, u, t, n)   X( u, t, n)

Proof of Theorem eclclwwlksn1

Step	Hyp	Ref	Expression
1		elqsecl 7801	. 2 |- ( B e. X -> ( B e. ( W /. .~ ) <-> E. x e. W B = { y | x .~ y } ) )
2		erclwwlksn.w	. . . . . . . . 9 |- W = ( N ClWWalksN G )
3		erclwwlksn.r	. . . . . . . . 9 |- .~ = { <. t , u >. | ( t e. W /\ u e. W /\ E. n e. ( 0 ... N ) t = ( u cyclShift n ) ) }
4	2, 3	erclwwlksnsym 26947	. . . . . . . 8 |- ( x .~ y -> y .~ x )
5	2, 3	erclwwlksnsym 26947	. . . . . . . 8 |- ( y .~ x -> x .~ y )
6	4, 5	impbii 199	. . . . . . 7 |- ( x .~ y <-> y .~ x )
7	6	a1i 11	. . . . . 6 |- ( ( B e. X /\ x e. W ) -> ( x .~ y <-> y .~ x ) )
8	7	abbidv 2741	. . . . 5 |- ( ( B e. X /\ x e. W ) -> { y | x .~ y } = { y | y .~ x } )
9		vex 3203	. . . . . . . 8 |- y e. _V
10		vex 3203	. . . . . . . 8 |- x e. _V
11	2, 3	erclwwlksneq 26944	. . . . . . . 8 |- ( ( y e. _V /\ x e. _V ) -> ( y .~ x <-> ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) )
12	9, 10, 11	mp2an 708	. . . . . . 7 |- ( y .~ x <-> ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) )
13	12	a1i 11	. . . . . 6 |- ( ( B e. X /\ x e. W ) -> ( y .~ x <-> ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) )
14	13	abbidv 2741	. . . . 5 |- ( ( B e. X /\ x e. W ) -> { y | y .~ x } = { y | ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) } )
15		3anan12 1051	. . . . . . . 8 |- ( ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) <-> ( x e. W /\ ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) )
16		ibar 525	. . . . . . . . . 10 |- ( x e. W -> ( ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) <-> ( x e. W /\ ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) ) )
17	16	bicomd 213	. . . . . . . . 9 |- ( x e. W -> ( ( x e. W /\ ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) <-> ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) )
18	17	adantl 482	. . . . . . . 8 |- ( ( B e. X /\ x e. W ) -> ( ( x e. W /\ ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) <-> ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) )
19	15, 18	syl5bb 272	. . . . . . 7 |- ( ( B e. X /\ x e. W ) -> ( ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) <-> ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) ) )
20	19	abbidv 2741	. . . . . 6 |- ( ( B e. X /\ x e. W ) -> { y | ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) } = { y | ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) } )
21		df-rab 2921	. . . . . 6 |- { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } = { y | ( y e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) }
22	20, 21	syl6eqr 2674	. . . . 5 |- ( ( B e. X /\ x e. W ) -> { y | ( y e. W /\ x e. W /\ E. n e. ( 0 ... N ) y = ( x cyclShift n ) ) } = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } )
23	8, 14, 22	3eqtrd 2660	. . . 4 |- ( ( B e. X /\ x e. W ) -> { y | x .~ y } = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } )
24	23	eqeq2d 2632	. . 3 |- ( ( B e. X /\ x e. W ) -> ( B = { y | x .~ y } <-> B = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) )
25	24	rexbidva 3049	. 2 |- ( B e. X -> ( E. x e. W B = { y | x .~ y } <-> E. x e. W B = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) )
26	1, 25	bitrd 268	1 |- ( B e. X -> ( B e. ( W /. .~ ) <-> E. x e. W B = { y e. W | E. n e. ( 0 ... N ) y = ( x cyclShift n ) } ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   {crab 2916   _Vcvv 3200   class class class wbr 4653   {copab 4712  (class class class)co 6650   /.cqs 7741   0cc0 9936   ...cfz 12326   cyclShift ccsh 13534   ClWWalksN cclwwlksn 26876

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  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-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-ec 7744  df-qs 7748  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-hash 13118  df-word 13299  df-concat 13301  df-substr 13303  df-csh 13535  df-clwwlks 26877  df-clwwlksn 26878

This theorem is referenced by:  eleclclwwlksn  26953  hashecclwwlksn1  26954  umgrhashecclwwlk  26955