Theorem wlkl1loop 16353

Description: A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)

Assertion

Ref	Expression
wlkl1loop	|- ( ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) )

Proof of Theorem wlkl1loop

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		wlkv 16321	. . . . 5 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2		simp3l 1052	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> Fun (iEdg ` G ) )
3		simp2 1025	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> F (Walks ` G ) P )
4		c0ex 8268	. . . . . . . . . . . . 13 |- 0 e. _V
5	4	snid 3720	. . . . . . . . . . . 12 |- 0 e. { 0 }
6		oveq2 6058	. . . . . . . . . . . . 13 |- ( (♯ ` F ) = 1 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 1 ) )
7		fzo01 10561	. . . . . . . . . . . . 13 |- ( 0..^ 1 ) = { 0 }
8	6, 7	eqtrdi 2281	. . . . . . . . . . . 12 |- ( (♯ ` F ) = 1 -> ( 0..^ (♯ ` F ) ) = { 0 } )
9	5, 8	eleqtrrid 2322	. . . . . . . . . . 11 |- ( (♯ ` F ) = 1 -> 0 e. ( 0..^ (♯ ` F ) ) )
10	9	ad2antrl 490	. . . . . . . . . 10 |- ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> 0 e. ( 0..^ (♯ ` F ) ) )
11	10	3ad2ant3 1047	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> 0 e. ( 0..^ (♯ ` F ) ) )
12		eqid 2232	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
13	12	iedginwlk 16352	. . . . . . . . 9 |- ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P /\ 0 e. ( 0..^ (♯ ` F ) ) ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) e. ran (iEdg ` G ) )
14	2, 3, 11, 13	syl3anc 1274	. . . . . . . 8 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) e. ran (iEdg ` G ) )
15		eqid 2232	. . . . . . . . . . 11 |- (Vtx ` G ) = (Vtx ` G )
16	15, 12	iswlkg 16324	. . . . . . . . . 10 |- ( G e. _V -> ( F (Walks ` G ) P <-> ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) ) )
17	8	raleqdv 2747	. . . . . . . . . . . . . . 15 |- ( (♯ ` F ) = 1 -> ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> A. k e. { 0 }if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) )
18		oveq1 6057	. . . . . . . . . . . . . . . . . 18 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
19		0p1e1 9351	. . . . . . . . . . . . . . . . . 18 |- ( 0 + 1 ) = 1
20	18, 19	eqtrdi 2281	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( k + 1 ) = 1 )
21		wkslem2 16316	. . . . . . . . . . . . . . . . 17 |- ( ( k = 0 /\ ( k + 1 ) = 1 ) -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
22	20, 21	mpdan 421	. . . . . . . . . . . . . . . 16 |- ( k = 0 -> (if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
23	4, 22	ralsn 3732	. . . . . . . . . . . . . . 15 |- ( A. k e. { 0 }if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
24	17, 23	bitrdi 196	. . . . . . . . . . . . . 14 |- ( (♯ ` F ) = 1 -> ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
25	24	ad2antrl 490	. . . . . . . . . . . . 13 |- ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) <-> if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
26		ifptru 998	. . . . . . . . . . . . . . . . 17 |- ( ( P ` 0 ) = ( P ` 1 ) -> (if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) <-> ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } ) )
27	26	biimpa 296	. . . . . . . . . . . . . . . 16 |- ( ( ( P ` 0 ) = ( P ` 1 ) /\ if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) -> ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } )
28	27	eqcomd 2238	. . . . . . . . . . . . . . 15 |- ( ( ( P ` 0 ) = ( P ` 1 ) /\ if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) )
29	28	ex 115	. . . . . . . . . . . . . 14 |- ( ( P ` 0 ) = ( P ` 1 ) -> (if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
30	29	ad2antll 491	. . . . . . . . . . . . 13 |- ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> (if- ( ( P ` 0 ) = ( P ` 1 ) , ( (iEdg ` G ) ` ( F ` 0 ) ) = { ( P ` 0 ) } , { ( P ` 0 ) , ( P ` 1 ) } C_ ( (iEdg ` G ) ` ( F ` 0 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
31	25, 30	sylbid 150	. . . . . . . . . . . 12 |- ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
32	31	com12 30	. . . . . . . . . . 11 |- ( A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) -> ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
33	32	3ad2ant3 1047	. . . . . . . . . 10 |- ( ( F e. Word dom (iEdg ` G ) /\ P : ( 0 ... (♯ ` F ) ) --> (Vtx ` G ) /\ A. k e. ( 0..^ (♯ ` F ) )if- ( ( P ` k ) = ( P ` ( k + 1 ) ) , ( (iEdg ` G ) ` ( F ` k ) ) = { ( P ` k ) } , { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ ( (iEdg ` G ) ` ( F ` k ) ) ) ) -> ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) )
34	16, 33	biimtrdi 163	. . . . . . . . 9 |- ( G e. _V -> ( F (Walks ` G ) P -> ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) ) ) )
35	34	3imp 1220	. . . . . . . 8 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> { ( P ` 0 ) } = ( (iEdg ` G ) ` ( F ` 0 ) ) )
36		edgvalg 16054	. . . . . . . . 9 |- ( G e. _V -> (Edg ` G ) = ran (iEdg ` G ) )
37	36	3ad2ant1 1045	. . . . . . . 8 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> (Edg ` G ) = ran (iEdg ` G ) )
38	14, 35, 37	3eltr4d 2316	. . . . . . 7 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) )
39	38	3exp 1229	. . . . . 6 |- ( G e. _V -> ( F (Walks ` G ) P -> ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) ) )
40	39	3ad2ant1 1045	. . . . 5 |- ( ( G e. _V /\ F e. _V /\ P e. _V ) -> ( F (Walks ` G ) P -> ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) ) )
41	1, 40	mpcom 36	. . . 4 |- ( F (Walks ` G ) P -> ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) )
42	41	expd 258	. . 3 |- ( F (Walks ` G ) P -> ( Fun (iEdg ` G ) -> ( ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) ) )
43	42	impcom 125	. 2 |- ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P ) -> ( ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) -> { ( P ` 0 ) } e. (Edg ` G ) ) )
44	43	imp 124	1 |- ( ( ( Fun (iEdg ` G ) /\ F (Walks ` G ) P ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> { ( P ` 0 ) } e. (Edg ` G ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105  if-wif 986   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   C_ wss 3211   {csn 3689   {cpr 3690   class class class wbr 4109   dom cdm 4749   ran crn 4750   Fun wfun 5346   -->wf 5348   ` cfv 5352  (class class class)co 6050   0cc0 8127   1c1 8128   + caddc 8130   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224  Vtxcvtx 16007  iEdgciedg 16008  Edgcedg 16052  Walkscwlks 16312

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-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  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-ifp 987  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-dom 6977  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-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-z 9578  df-dec 9710  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-edg 16053  df-wlks 16313

This theorem is referenced by: (None)