Theorem clwwlkn1loopb 16415

Description: A word represents a closed walk of length 1 iff this word is a singleton word consisting of a vertex with an attached loop. (Contributed by AV, 11-Feb-2022.)

Assertion

Ref	Expression
clwwlkn1loopb	|- ( W e. ( 1 ClWWalksN G ) <-> E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) )

Distinct variable groups:   v, G   v, W

Proof of Theorem clwwlkn1loopb

Step	Hyp	Ref	Expression
1		clwwlkn1 16413	. 2 |- ( W e. ( 1 ClWWalksN G ) <-> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )
2		wrdl1exs1 11317	. . . . . 6 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = 1 ) -> E. v e. (Vtx ` G ) W = <" v "> )
3		fveq1 5669	. . . . . . . . . . . . . . 15 |- ( W = <" v "> -> ( W ` 0 ) = ( <" v "> ` 0 ) )
4		s1fv 11314	. . . . . . . . . . . . . . 15 |- ( v e. (Vtx ` G ) -> ( <" v "> ` 0 ) = v )
5	3, 4	sylan9eq 2285	. . . . . . . . . . . . . 14 |- ( ( W = <" v "> /\ v e. (Vtx ` G ) ) -> ( W ` 0 ) = v )
6	5	sneqd 3702	. . . . . . . . . . . . 13 |- ( ( W = <" v "> /\ v e. (Vtx ` G ) ) -> { ( W ` 0 ) } = { v } )
7	6	eleq1d 2301	. . . . . . . . . . . 12 |- ( ( W = <" v "> /\ v e. (Vtx ` G ) ) -> ( { ( W ` 0 ) } e. (Edg ` G ) <-> { v } e. (Edg ` G ) ) )
8	7	biimpd 144	. . . . . . . . . . 11 |- ( ( W = <" v "> /\ v e. (Vtx ` G ) ) -> ( { ( W ` 0 ) } e. (Edg ` G ) -> { v } e. (Edg ` G ) ) )
9	8	ex 115	. . . . . . . . . 10 |- ( W = <" v "> -> ( v e. (Vtx ` G ) -> ( { ( W ` 0 ) } e. (Edg ` G ) -> { v } e. (Edg ` G ) ) ) )
10	9	com13 80	. . . . . . . . 9 |- ( { ( W ` 0 ) } e. (Edg ` G ) -> ( v e. (Vtx ` G ) -> ( W = <" v "> -> { v } e. (Edg ` G ) ) ) )
11	10	imp 124	. . . . . . . 8 |- ( ( { ( W ` 0 ) } e. (Edg ` G ) /\ v e. (Vtx ` G ) ) -> ( W = <" v "> -> { v } e. (Edg ` G ) ) )
12	11	ancld 325	. . . . . . 7 |- ( ( { ( W ` 0 ) } e. (Edg ` G ) /\ v e. (Vtx ` G ) ) -> ( W = <" v "> -> ( W = <" v "> /\ { v } e. (Edg ` G ) ) ) )
13	12	reximdva 2644	. . . . . 6 |- ( { ( W ` 0 ) } e. (Edg ` G ) -> ( E. v e. (Vtx ` G ) W = <" v "> -> E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) ) )
14	2, 13	syl5com 29	. . . . 5 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = 1 ) -> ( { ( W ` 0 ) } e. (Edg ` G ) -> E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) ) )
15	14	expcom 116	. . . 4 |- ( (♯ ` W ) = 1 -> ( W e. Word (Vtx ` G ) -> ( { ( W ` 0 ) } e. (Edg ` G ) -> E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) ) ) )
16	15	3imp 1220	. . 3 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) -> E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) )
17		s1leng 11312	. . . . . . . 8 |- ( v e. (Vtx ` G ) -> (♯ ` <" v "> ) = 1 )
18	17	adantr 276	. . . . . . 7 |- ( ( v e. (Vtx ` G ) /\ { v } e. (Edg ` G ) ) -> (♯ ` <" v "> ) = 1 )
19		s1cl 11309	. . . . . . . 8 |- ( v e. (Vtx ` G ) -> <" v "> e. Word (Vtx ` G ) )
20	19	adantr 276	. . . . . . 7 |- ( ( v e. (Vtx ` G ) /\ { v } e. (Edg ` G ) ) -> <" v "> e. Word (Vtx ` G ) )
21	4	eqcomd 2238	. . . . . . . . . . 11 |- ( v e. (Vtx ` G ) -> v = ( <" v "> ` 0 ) )
22	21	sneqd 3702	. . . . . . . . . 10 |- ( v e. (Vtx ` G ) -> { v } = { ( <" v "> ` 0 ) } )
23	22	eleq1d 2301	. . . . . . . . 9 |- ( v e. (Vtx ` G ) -> ( { v } e. (Edg ` G ) <-> { ( <" v "> ` 0 ) } e. (Edg ` G ) ) )
24	23	biimpd 144	. . . . . . . 8 |- ( v e. (Vtx ` G ) -> ( { v } e. (Edg ` G ) -> { ( <" v "> ` 0 ) } e. (Edg ` G ) ) )
25	24	imp 124	. . . . . . 7 |- ( ( v e. (Vtx ` G ) /\ { v } e. (Edg ` G ) ) -> { ( <" v "> ` 0 ) } e. (Edg ` G ) )
26	18, 20, 25	3jca 1204	. . . . . 6 |- ( ( v e. (Vtx ` G ) /\ { v } e. (Edg ` G ) ) -> ( (♯ ` <" v "> ) = 1 /\ <" v "> e. Word (Vtx ` G ) /\ { ( <" v "> ` 0 ) } e. (Edg ` G ) ) )
27	26	adantrl 478	. . . . 5 |- ( ( v e. (Vtx ` G ) /\ ( W = <" v "> /\ { v } e. (Edg ` G ) ) ) -> ( (♯ ` <" v "> ) = 1 /\ <" v "> e. Word (Vtx ` G ) /\ { ( <" v "> ` 0 ) } e. (Edg ` G ) ) )
28		fveqeq2 5679	. . . . . . 7 |- ( W = <" v "> -> ( (♯ ` W ) = 1 <-> (♯ ` <" v "> ) = 1 ) )
29		eleq1 2295	. . . . . . 7 |- ( W = <" v "> -> ( W e. Word (Vtx ` G ) <-> <" v "> e. Word (Vtx ` G ) ) )
30	3	sneqd 3702	. . . . . . . 8 |- ( W = <" v "> -> { ( W ` 0 ) } = { ( <" v "> ` 0 ) } )
31	30	eleq1d 2301	. . . . . . 7 |- ( W = <" v "> -> ( { ( W ` 0 ) } e. (Edg ` G ) <-> { ( <" v "> ` 0 ) } e. (Edg ` G ) ) )
32	28, 29, 31	3anbi123d 1349	. . . . . 6 |- ( W = <" v "> -> ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) <-> ( (♯ ` <" v "> ) = 1 /\ <" v "> e. Word (Vtx ` G ) /\ { ( <" v "> ` 0 ) } e. (Edg ` G ) ) ) )
33	32	ad2antrl 490	. . . . 5 |- ( ( v e. (Vtx ` G ) /\ ( W = <" v "> /\ { v } e. (Edg ` G ) ) ) -> ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) <-> ( (♯ ` <" v "> ) = 1 /\ <" v "> e. Word (Vtx ` G ) /\ { ( <" v "> ` 0 ) } e. (Edg ` G ) ) ) )
34	27, 33	mpbird 167	. . . 4 |- ( ( v e. (Vtx ` G ) /\ ( W = <" v "> /\ { v } e. (Edg ` G ) ) ) -> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )
35	34	rexlimiva 2655	. . 3 |- ( E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) -> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )
36	16, 35	impbii 126	. 2 |- ( ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) <-> E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) )
37	1, 36	bitri 184	1 |- ( W e. ( 1 ClWWalksN G ) <-> E. v e. (Vtx ` G ) ( W = <" v "> /\ { v } e. (Edg ` G ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   E.wrex 2521   {csn 3689   ` cfv 5352  (class class class)co 6050   0cc0 8127   1c1 8128  ♯chash 11138  Word cword 11224   <"cs1 11303  Vtxcvtx 16007  Edgcedg 16052   ClWWalksN cclwwlkn 16398

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-mulrcl 8226  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-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

This theorem depends on definitions:  df-bi 117  df-dc 843  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-reap 8849  df-ap 8856  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-lsw 11270  df-s1 11304  df-ndx 13215  df-slot 13216  df-base 13218  df-vtx 16009  df-clwwlk 16387  df-clwwlkn 16399

This theorem is referenced by: (None)