Theorem wlkl1loop 16208

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 16176	. . . . 5 |- ( F (Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )
2		simp3l 1051	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> Fun (iEdg ` G ) )
3		simp2 1024	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> F (Walks ` G ) P )
4		c0ex 8172	. . . . . . . . . . . . 13 |- 0 e. _V
5	4	snid 3700	. . . . . . . . . . . 12 |- 0 e. { 0 }
6		oveq2 6025	. . . . . . . . . . . . 13 |- ( (♯ ` F ) = 1 -> ( 0..^ (♯ ` F ) ) = ( 0..^ 1 ) )
7		fzo01 10460	. . . . . . . . . . . . 13 |- ( 0..^ 1 ) = { 0 }
8	6, 7	eqtrdi 2280	. . . . . . . . . . . 12 |- ( (♯ ` F ) = 1 -> ( 0..^ (♯ ` F ) ) = { 0 } )
9	5, 8	eleqtrrid 2321	. . . . . . . . . . 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 1046	. . . . . . . . 9 |- ( ( G e. _V /\ F (Walks ` G ) P /\ ( Fun (iEdg ` G ) /\ ( (♯ ` F ) = 1 /\ ( P ` 0 ) = ( P ` 1 ) ) ) ) -> 0 e. ( 0..^ (♯ ` F ) ) )
12		eqid 2231	. . . . . . . . . 10 |- (iEdg ` G ) = (iEdg ` G )
13	12	iedginwlk 16207	. . . . . . . . 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 1273	. . . . . . . 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 2231	. . . . . . . . . . 11 |- (Vtx ` G ) = (Vtx ` G )
16	15, 12	iswlkg 16179	. . . . . . . . . 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 2736	. . . . . . . . . . . . . . 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 6024	. . . . . . . . . . . . . . . . . 18 |- ( k = 0 -> ( k + 1 ) = ( 0 + 1 ) )
19		0p1e1 9256	. . . . . . . . . . . . . . . . . 18 |- ( 0 + 1 ) = 1
20	18, 19	eqtrdi 2280	. . . . . . . . . . . . . . . . 17 |- ( k = 0 -> ( k + 1 ) = 1 )
21		wkslem2 16171	. . . . . . . . . . . . . . . . 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 3712	. . . . . . . . . . . . . . 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 997	. . . . . . . . . . . . . . . . 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 2237	. . . . . . . . . . . . . . 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 1046	. . . . . . . . . 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 1219	. . . . . . . 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 15909	. . . . . . . . 9 |- ( G e. _V -> (Edg ` G ) = ran (iEdg ` G ) )
37	36	3ad2ant1 1044	. . . . . . . 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 2315	. . . . . . 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 1228	. . . . . 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 1044	. . . . 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 985   /\ w3a 1004   = wceq 1397   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   {csn 3669   {cpr 3670   class class class wbr 4088   dom cdm 4725   ran crn 4726   Fun wfun 5320   -->wf 5322   ` cfv 5326  (class class class)co 6017   0cc0 8031   1c1 8032   + caddc 8034   ...cfz 10242  ..^cfzo 10376  ♯chash 11036  Word cword 11112  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  Walkscwlks 16167

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-ifp 986  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-1o 6581  df-er 6701  df-map 6818  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-z 9479  df-dec 9611  df-uz 9755  df-fz 10243  df-fzo 10377  df-ihash 11037  df-word 11113  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-wlks 16168

This theorem is referenced by: (None)