Theorem clwwlkn1loopb 16361

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 16359	. 2 |- ( W e. ( 1 ClWWalksN G ) <-> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )
2		wrdl1exs1 11272	. . . . . 6 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = 1 ) -> E. v e. (Vtx ` G ) W = <" v "> )
3		fveq1 5647	. . . . . . . . . . . . . . 15 |- ( W = <" v "> -> ( W ` 0 ) = ( <" v "> ` 0 ) )
4		s1fv 11269	. . . . . . . . . . . . . . 15 |- ( v e. (Vtx ` G ) -> ( <" v "> ` 0 ) = v )
5	3, 4	sylan9eq 2284	. . . . . . . . . . . . . 14 |- ( ( W = <" v "> /\ v e. (Vtx ` G ) ) -> ( W ` 0 ) = v )
6	5	sneqd 3686	. . . . . . . . . . . . 13 |- ( ( W = <" v "> /\ v e. (Vtx ` G ) ) -> { ( W ` 0 ) } = { v } )
7	6	eleq1d 2300	. . . . . . . . . . . 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 2635	. . . . . 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 11267	. . . . . . . 8 |- ( v e. (Vtx ` G ) -> (♯ ` <" v "> ) = 1 )
18	17	adantr 276	. . . . . . 7 |- ( ( v e. (Vtx ` G ) /\ { v } e. (Edg ` G ) ) -> (♯ ` <" v "> ) = 1 )
19		s1cl 11264	. . . . . . . 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 2237	. . . . . . . . . . 11 |- ( v e. (Vtx ` G ) -> v = ( <" v "> ` 0 ) )
22	21	sneqd 3686	. . . . . . . . . 10 |- ( v e. (Vtx ` G ) -> { v } = { ( <" v "> ` 0 ) } )
23	22	eleq1d 2300	. . . . . . . . 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 5657	. . . . . . 7 |- ( W = <" v "> -> ( (♯ ` W ) = 1 <-> (♯ ` <" v "> ) = 1 ) )
29		eleq1 2294	. . . . . . 7 |- ( W = <" v "> -> ( W e. Word (Vtx ` G ) <-> <" v "> e. Word (Vtx ` G ) ) )
30	3	sneqd 3686	. . . . . . . 8 |- ( W = <" v "> -> { ( W ` 0 ) } = { ( <" v "> ` 0 ) } )
31	30	eleq1d 2300	. . . . . . 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 2646	. . 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 2202   E.wrex 2512   {csn 3673   ` cfv 5333  (class class class)co 6028   0cc0 8092   1c1 8093  ♯chash 11100  Word cword 11179   <"cs1 11258  Vtxcvtx 15953  Edgcedg 15998   ClWWalksN cclwwlkn 16344

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-map 6862  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-fz 10306  df-fzo 10440  df-ihash 11101  df-word 11180  df-lsw 11225  df-s1 11259  df-ndx 13165  df-slot 13166  df-base 13168  df-vtx 15955  df-clwwlk 16333  df-clwwlkn 16345

This theorem is referenced by: (None)