Theorem clwwlkg 16388

Description: The set of closed walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)

Hypotheses

Ref	Expression
clwwlk.v	|- V = (Vtx ` G )
clwwlk.e	|- E = (Edg ` G )

Assertion

Ref	Expression
clwwlkg	|- ( G e. W -> (ClWWalks ` G ) = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { (lastS ` w ) , ( w ` 0 ) } e. E ) } )

Distinct variable groups:   i, G, w   w, V

Allowed substitution hints:   E( w, i)   V( i)   W( w, i)

Proof of Theorem clwwlkg

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clwwlk 16387	. 2 |- ClWWalks = ( g e. _V |-> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) } )
2		fveq2 5670	. . . . 5 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
3		clwwlk.v	. . . . 5 |- V = (Vtx ` G )
4	2, 3	eqtr4di 2283	. . . 4 |- ( g = G -> (Vtx ` g ) = V )
5		wrdeq 11246	. . . 4 |- ( (Vtx ` g ) = V -> Word (Vtx ` g ) = Word V )
6	4, 5	syl 14	. . 3 |- ( g = G -> Word (Vtx ` g ) = Word V )
7		fveq2 5670	. . . . . . 7 |- ( g = G -> (Edg ` g ) = (Edg ` G ) )
8		clwwlk.e	. . . . . . 7 |- E = (Edg ` G )
9	7, 8	eqtr4di 2283	. . . . . 6 |- ( g = G -> (Edg ` g ) = E )
10	9	eleq2d 2302	. . . . 5 |- ( g = G -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) <-> { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) )
11	10	ralbidv 2542	. . . 4 |- ( g = G -> ( A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) <-> A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) )
12	9	eleq2d 2302	. . . 4 |- ( g = G -> ( { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) <-> { (lastS ` w ) , ( w ` 0 ) } e. E ) )
13	11, 12	3anbi23d 1352	. . 3 |- ( g = G -> ( ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) <-> ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { (lastS ` w ) , ( w ` 0 ) } e. E ) ) )
14	6, 13	rabeqbidv 2808	. 2 |- ( g = G -> { w e. Word (Vtx ` g ) | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) /\ { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) ) } = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { (lastS ` w ) , ( w ` 0 ) } e. E ) } )
15		elex 2825	. 2 |- ( G e. W -> G e. _V )
16		vtxex 16013	. . . 4 |- ( G e. W -> (Vtx ` G ) e. _V )
17	3, 16	eqeltrid 2319	. . 3 |- ( G e. W -> V e. _V )
18		wrdexg 11235	. . 3 |- ( V e. _V -> Word V e. _V )
19		rabexg 4255	. . 3 |- (Word V e. _V -> { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { (lastS ` w ) , ( w ` 0 ) } e. E ) } e. _V )
20	17, 18, 19	3syl 17	. 2 |- ( G e. W -> { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { (lastS ` w ) , ( w ` 0 ) } e. E ) } e. _V )
21	1, 14, 15, 20	fvmptd3 5771	1 |- ( G e. W -> (ClWWalks ` G ) = { w e. Word V | ( w =/= (/) /\ A. i e. ( 0..^ ( (♯ ` w ) - 1 ) ) { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E /\ { (lastS ` w ) , ( w ` 0 ) } e. E ) } )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2203   =/= wne 2412   A.wral 2520   {crab 2524   _Vcvv 2813   (/)c0 3508   {cpr 3690   ` cfv 5352  (class class class)co 6050   0cc0 8127   1c1 8128   + caddc 8130   - cmin 8444  ..^cfzo 10476  ♯chash 11138  Word cword 11224  lastSclsw 11269  Vtxcvtx 16007  Edgcedg 16052  ClWWalkscclwwlk 16386

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-addass 8229  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-map 6884  df-en 6976  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-word 11225  df-ndx 13215  df-slot 13216  df-base 13218  df-vtx 16009  df-clwwlk 16387

This theorem is referenced by:  isclwwlk  16389