Theorem clwlksfclwwlk2wrd 26958

Description: The second component of a closed walk is a word over the "vertices". (Contributed by Alexander van der Vekens, 25-Jun-2018.) (Revised by AV, 2-May-2021.)

Hypotheses

Ref	Expression
clwlksfclwwlk.1	|- A = ( 1st ` c )
clwlksfclwwlk.2	|- B = ( 2nd ` c )
clwlksfclwwlk.c	|- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
clwlksfclwwlk.f	|- F = ( c e. C |-> ( B substr <. 0 , ( # ` A ) >. ) )

Assertion

Ref	Expression
clwlksfclwwlk2wrd	|- ( c e. C -> B e. Word (Vtx ` G ) )

Distinct variable group:   G, c

Allowed substitution hints:   A( c)   B( c)   C( c)   F( c)   N( c)

Proof of Theorem clwlksfclwwlk2wrd

Dummy variable i is distinct from all other variables.

Step	Hyp	Ref	Expression
1		clwlksfclwwlk.c	. . 3 |- C = { c e. (ClWalks ` G ) | ( # ` A ) = N }
2	1	rabeq2i 3197	. 2 |- ( c e. C <-> ( c e. (ClWalks ` G ) /\ ( # ` A ) = N ) )
3		eqid 2622	. . . . 5 |- (Vtx ` G ) = (Vtx ` G )
4		eqid 2622	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
5		clwlksfclwwlk.1	. . . . 5 |- A = ( 1st ` c )
6		clwlksfclwwlk.2	. . . . 5 |- B = ( 2nd ` c )
7	3, 4, 5, 6	clwlkcompim 26676	. . . 4 |- ( c e. (ClWalks ` G ) -> ( ( A e. Word dom (iEdg ` G ) /\ B : ( 0 ... ( # ` A ) ) --> (Vtx ` G ) ) /\ ( A. i e. ( 0..^ ( # ` A ) )if- ( ( B ` i ) = ( B ` ( i + 1 ) ) , ( (iEdg ` G ) ` ( A ` i ) ) = { ( B ` i ) } , { ( B ` i ) , ( B ` ( i + 1 ) ) } C_ ( (iEdg ` G ) ` ( A ` i ) ) ) /\ ( B ` 0 ) = ( B ` ( # ` A ) ) ) ) )
8		lencl 13324	. . . . . 6 |- ( A e. Word dom (iEdg ` G ) -> ( # ` A ) e. NN0 )
9		nn0z 11400	. . . . . . . . . 10 |- ( ( # ` A ) e. NN0 -> ( # ` A ) e. ZZ )
10		fzval3 12536	. . . . . . . . . 10 |- ( ( # ` A ) e. ZZ -> ( 0 ... ( # ` A ) ) = ( 0..^ ( ( # ` A ) + 1 ) ) )
11	9, 10	syl 17	. . . . . . . . 9 |- ( ( # ` A ) e. NN0 -> ( 0 ... ( # ` A ) ) = ( 0..^ ( ( # ` A ) + 1 ) ) )
12	11	feq2d 6031	. . . . . . . 8 |- ( ( # ` A ) e. NN0 -> ( B : ( 0 ... ( # ` A ) ) --> (Vtx ` G ) <-> B : ( 0..^ ( ( # ` A ) + 1 ) ) --> (Vtx ` G ) ) )
13	12	biimpa 501	. . . . . . 7 |- ( ( ( # ` A ) e. NN0 /\ B : ( 0 ... ( # ` A ) ) --> (Vtx ` G ) ) -> B : ( 0..^ ( ( # ` A ) + 1 ) ) --> (Vtx ` G ) )
14		iswrdi 13309	. . . . . . 7 |- ( B : ( 0..^ ( ( # ` A ) + 1 ) ) --> (Vtx ` G ) -> B e. Word (Vtx ` G ) )
15	13, 14	syl 17	. . . . . 6 |- ( ( ( # ` A ) e. NN0 /\ B : ( 0 ... ( # ` A ) ) --> (Vtx ` G ) ) -> B e. Word (Vtx ` G ) )
16	8, 15	sylan 488	. . . . 5 |- ( ( A e. Word dom (iEdg ` G ) /\ B : ( 0 ... ( # ` A ) ) --> (Vtx ` G ) ) -> B e. Word (Vtx ` G ) )
17	16	adantr 481	. . . 4 |- ( ( ( A e. Word dom (iEdg ` G ) /\ B : ( 0 ... ( # ` A ) ) --> (Vtx ` G ) ) /\ ( A. i e. ( 0..^ ( # ` A ) )if- ( ( B ` i ) = ( B ` ( i + 1 ) ) , ( (iEdg ` G ) ` ( A ` i ) ) = { ( B ` i ) } , { ( B ` i ) , ( B ` ( i + 1 ) ) } C_ ( (iEdg ` G ) ` ( A ` i ) ) ) /\ ( B ` 0 ) = ( B ` ( # ` A ) ) ) ) -> B e. Word (Vtx ` G ) )
18	7, 17	syl 17	. . 3 |- ( c e. (ClWalks ` G ) -> B e. Word (Vtx ` G ) )
19	18	adantr 481	. 2 |- ( ( c e. (ClWalks ` G ) /\ ( # ` A ) = N ) -> B e. Word (Vtx ` G ) )
20	2, 19	sylbi 207	1 |- ( c e. C -> B e. Word (Vtx ` G ) )

Syntax hints:   -> wi 4   /\ wa 384  if-wif 1012   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   1stc1st 7166   2ndc2nd 7167   0cc0 9936   1c1 9937   + caddc 9939   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   substr csubstr 13295  Vtxcvtx 25874  iEdgciedg 25875  ClWalkscclwlks 26666

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013  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-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-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-wlks 26495  df-clwlks 26667

This theorem is referenced by:  clwlksfclwwlk2sswd  26961  clwlksfclwwlk  26962