Theorem wlkl1loop 16069

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 16038	. . . . 5 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2		simp3l 1049	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> Fun (iEdg ` G ) )
3		simp2 1022	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> F (Walks ` G ) P )
4		c0ex 8140	. . . . . . . . . . . . 13 |- 0 e. _V
5	4	snid 3697	. . . . . . . . . . . 12 |- 0 e. { 0 }
6		oveq2 6009	. . . . . . . . . . . . 13 |- ( (♯ ` F ) = 1 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 1 ) )
7		fzo01 10422	. . . . . . . . . . . . 13 |- ( 0..^ 1 ) = { 0 }
8	6, 7	eqtrdi 2278	. . . . . . . . . . . 12 |- ( (♯ ` F ) = 1 -> ( 0..^ (♯ ` F ) ) = { 0 } )
9	5, 8	eleqtrrid 2319	. . . . . . . . . . 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 1044	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> 0 e. ( 0..^ (♯ ` F ) ) )
12		eqid 2229	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
13	12	iedginwlk 16068	. . . . . . . . 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 1271	. . . . . . . 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 2229	. . . . . . . . . . 11 |- (Vtx ` G ) = (Vtx ` G )
16	15, 12	iswlkg 16041	. . . . . . . . . 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 2734	. . . . . . . . . . . . . . 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 6008	. . . . . . . . . . . . . . . . . 18 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
19		0p1e1 9224	. . . . . . . . . . . . . . . . . 18 |- ( 0 + 1 ) = 1
20	18, 19	eqtrdi 2278	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( k + 1 ) = 1 )
21		wkslem2 16034	. . . . . . . . . . . . . . . . 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 3709	. . . . . . . . . . . . . . 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 995	. . . . . . . . . . . . . . . . 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 2235	. . . . . . . . . . . . . . 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 1044	. . . . . . . . . 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 1217	. . . . . . . 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 15860	. . . . . . . . 9 |- ( G e. _V -> (Edg ` G ) = ran (iEdg ` G ) )
37	36	3ad2ant1 1042	. . . . . . . 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 2313	. . . . . . 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 1226	. . . . . 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 1042	. . . . 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 983   /\ w3a 1002   = wceq 1395   e. wcel 2200   A.wral 2508   _Vcvv 2799   C_ wss 3197   {csn 3666   {cpr 3667   class class class wbr 4083   dom cdm 4719   ran crn 4720   Fun wfun 5312   -->wf 5314   ` cfv 5318  (class class class)co 6001   0cc0 7999   1c1 8000   + caddc 8002   ...cfz 10204  ..^cfzo 10338  ♯chash 10997  Word cword 11071  Vtxcvtx 15813  iEdgciedg 15814  Edgcedg 15858  Walkscwlks 16030

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-mulcom 8100  ax-addass 8101  ax-mulass 8102  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-1rid 8106  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-apti 8114  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984  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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-1st 6286  df-2nd 6287  df-recs 6451  df-frec 6537  df-1o 6562  df-er 6680  df-map 6797  df-en 6888  df-dom 6889  df-fin 6890  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-2 9169  df-3 9170  df-4 9171  df-5 9172  df-6 9173  df-7 9174  df-8 9175  df-9 9176  df-n0 9370  df-z 9447  df-dec 9579  df-uz 9723  df-fz 10205  df-fzo 10339  df-ihash 10998  df-word 11072  df-ndx 13035  df-slot 13036  df-base 13038  df-edgf 15806  df-vtx 15815  df-iedg 15816  df-edg 15859  df-wlks 16031

This theorem is referenced by: (None)