Theorem clwwlkg 16188

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 16187	. 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 5635	. . . . 5 |- ( g = G -> (Vtx ` g ) = (Vtx ` G ) )
3		clwwlk.v	. . . . 5 |- V = (Vtx ` G )
4	2, 3	eqtr4di 2280	. . . 4 |- ( g = G -> (Vtx ` g ) = V )
5		wrdeq 11125	. . . 4 |- ( (Vtx ` g ) = V -> Word (Vtx ` g ) = Word V )
6	4, 5	syl 14	. . 3 |- ( g = G -> Word (Vtx ` g ) = Word V )
7		fveq2 5635	. . . . . . 7 |- ( g = G -> (Edg ` g ) = (Edg ` G ) )
8		clwwlk.e	. . . . . . 7 |- E = (Edg ` G )
9	7, 8	eqtr4di 2280	. . . . . 6 |- ( g = G -> (Edg ` g ) = E )
10	9	eleq2d 2299	. . . . 5 |- ( g = G -> ( { ( w ` i ) , ( w ` ( i + 1 ) ) } e. (Edg ` g ) <-> { ( w ` i ) , ( w ` ( i + 1 ) ) } e. E ) )
11	10	ralbidv 2530	. . . 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 2299	. . . 4 |- ( g = G -> ( { (lastS ` w ) , ( w ` 0 ) } e. (Edg ` g ) <-> { (lastS ` w ) , ( w ` 0 ) } e. E ) )
13	11, 12	3anbi23d 1349	. . 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 2795	. 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 2812	. 2 |- ( G e. W -> G e. _V )
16		vtxex 15859	. . . 4 |- ( G e. W -> (Vtx ` G ) e. _V )
17	3, 16	eqeltrid 2316	. . 3 |- ( G e. W -> V e. _V )
18		wrdexg 11114	. . 3 |- ( V e. _V -> Word V e. _V )
19		rabexg 4231	. . 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 5736	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 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   A.wral 2508   {crab 2512   _Vcvv 2800   (/)c0 3492   {cpr 3668   ` cfv 5324  (class class class)co 6013   0cc0 8022   1c1 8023   + caddc 8025   - cmin 8340  ..^cfzo 10367  ♯chash 11027  Word cword 11103  lastSclsw 11148  Vtxcvtx 15853  Edgcedg 15898  ClWWalkscclwwlk 16186

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-addcom 8122  ax-addass 8124  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-0id 8130  ax-rnegex 8131  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-map 6814  df-en 6905  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-fzo 10368  df-word 11104  df-ndx 13075  df-slot 13076  df-base 13078  df-vtx 15855  df-clwwlk 16187

This theorem is referenced by:  isclwwlk  16189