Theorem wlkl1loop 16599

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 16567	. . . . 5 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2		simp3l 1056	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> Fun (iEdg ` G ) )
3		simp2 1029	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> F (Walks ` G ) P )
4		c0ex 8320	. . . . . . . . . . . . 13 |- 0 e. _V
5	4	snid 3740	. . . . . . . . . . . 12 |- 0 e. { 0 }
6		oveq2 6093	. . . . . . . . . . . . 13 |- ( (♯ ` F ) = 1 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 1 ) )
7		fzo01 10634	. . . . . . . . . . . . 13 |- ( 0..^ 1 ) = { 0 }
8	6, 7	eqtrdi 2287	. . . . . . . . . . . 12 |- ( (♯ ` F ) = 1 -> ( 0..^ (♯ ` F ) ) = { 0 } )
9	5, 8	eleqtrrid 2328	. . . . . . . . . . 11 |- ( (♯ ` F ) = 1 -> 0 e. ( 0..^ (♯ ` F ) ) )
10	9	ad2antrl 494	. . . . . . . . . 10 |- ( ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) -> 0 e. ( 0..^ (♯ ` F ) ) )
11	10	3ad2ant3 1051	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> 0 e. ( 0..^ (♯ ` F ) ) )
12		eqid 2238	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
13	12	iedginwlk 16598	. . . . . . . . 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 1278	. . . . . . . 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 2238	. . . . . . . . . . 11 |- (Vtx ` G ) = (Vtx ` G )
16	15, 12	iswlkg 16570	. . . . . . . . . 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 2755	. . . . . . . . . . . . . . 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 6092	. . . . . . . . . . . . . . . . . 18 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
19		0p1e1 9418	. . . . . . . . . . . . . . . . . 18 |- ( 0 + 1 ) = 1
20	18, 19	eqtrdi 2287	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( k + 1 ) = 1 )
21		wkslem2 16562	. . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . 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 3752	. . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . 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 1002	. . . . . . . . . . . . . . . . 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 2244	. . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 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 1051	. . . . . . . . . 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 1224	. . . . . . . 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 16300	. . . . . . . . 9 |- ( G e. _V -> (Edg ` G ) = ran (iEdg ` G ) )
37	36	3ad2ant1 1049	. . . . . . . 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 2322	. . . . . . 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 1233	. . . . . 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 1049	. . . . 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105  if-wif 990   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   {csn 3709   {cpr 3710   class class class wbr 4130   dom cdm 4774   ran crn 4775   Fun wfun 5371   -->wf 5373   ` cfv 5377  (class class class)co 6085   0cc0 8179   1c1 8180   + caddc 8182   ...cfz 10411  ..^cfzo 10549  ♯chash 11214  Word cword 11304  Vtxcvtx 16253  iEdgciedg 16254  Edgcedg 16298  Walkscwlks 16558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-ifp 991  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-map 6924  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-z 9645  df-dec 9778  df-uz 9922  df-fz 10412  df-fzo 10550  df-ihash 11215  df-word 11305  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256  df-edg 16299  df-wlks 16559

This theorem is used by: (None)