Theorem clwwlkn1loopb 16575

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 16573	. 2 |- ( W e. ( 1 ClWWalksN G ) <-> ( (♯ ` W ) = 1 /\ W e. Word (Vtx ` G ) /\ { ( W ` 0 ) } e. (Edg ` G ) ) )
2		wrdl1exs1 11375	. . . . . 6 |- ( ( W e. Word (Vtx ` G ) /\ (♯ ` W ) = 1 ) -> E. v e. (Vtx ` G ) W = <" v "> )
3		fveq1 5689	. . . . . . . . . . . . . . 15 |- ( W = <" v "> -> ( W ` 0 ) = ( <" v "> ` 0 ) )
4		s1fv 11372	. . . . . . . . . . . . . . 15 |- ( v e. (Vtx ` G ) -> ( <" v "> ` 0 ) = v )
5	3, 4	sylan9eq 2291	. . . . . . . . . . . . . 14 |- ( ( W = <" v "> /\ v e. (Vtx ` G ) ) -> ( W ` 0 ) = v )
6	5	sneqd 3718	. . . . . . . . . . . . 13 |- ( ( W = <" v "> /\ v e. (Vtx ` G ) ) -> { ( W ` 0 ) } = { v } )
7	6	eleq1d 2307	. . . . . . . . . . . 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 2652	. . . . . 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 1224	. . 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 11370	. . . . . . . 8 |- ( v e. (Vtx ` G ) -> (♯ ` <" v "> ) = 1 )
18	17	adantr 276	. . . . . . 7 |- ( ( v e. (Vtx ` G ) /\ { v } e. (Edg ` G ) ) -> (♯ ` <" v "> ) = 1 )
19		s1cl 11367	. . . . . . . 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 2244	. . . . . . . . . . 11 |- ( v e. (Vtx ` G ) -> v = ( <" v "> ` 0 ) )
22	21	sneqd 3718	. . . . . . . . . 10 |- ( v e. (Vtx ` G ) -> { v } = { ( <" v "> ` 0 ) } )
23	22	eleq1d 2307	. . . . . . . . 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 1208	. . . . . 6 |- ( ( v e. (Vtx ` G ) /\ { v } e. (Edg ` G ) ) -> ( (♯ ` <" v "> ) = 1 /\ <" v "> e. Word (Vtx ` G ) /\ { ( <" v "> ` 0 ) } e. (Edg ` G ) ) )
27	26	adantrl 482	. . . . 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 5699	. . . . . . 7 |- ( W = <" v "> -> ( (♯ ` W ) = 1 <-> (♯ ` <" v "> ) = 1 ) )
29		eleq1 2301	. . . . . . 7 |- ( W = <" v "> -> ( W e. Word (Vtx ` G ) <-> <" v "> e. Word (Vtx ` G ) ) )
30	3	sneqd 3718	. . . . . . . 8 |- ( W = <" v "> -> { ( W ` 0 ) } = { ( <" v "> ` 0 ) } )
31	30	eleq1d 2307	. . . . . . 7 |- ( W = <" v "> -> ( { ( W ` 0 ) } e. (Edg ` G ) <-> { ( <" v "> ` 0 ) } e. (Edg ` G ) ) )
32	28, 29, 31	3anbi123d 1353	. . . . . 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 494	. . . . 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 2663	. . 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 1009   = wceq 1402   e. wcel 2209   E.wrex 2529   {csn 3705   ` cfv 5372  (class class class)co 6075   0cc0 8169   1c1 8170  ♯chash 11192  Word cword 11282   <"cs1 11361  Vtxcvtx 16167  Edgcedg 16212   ClWWalksN cclwwlkn 16558

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

This theorem depends on definitions:  df-bi 117  df-dc 847  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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-map 6914  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-lsw 11328  df-s1 11362  df-ndx 13333  df-slot 13334  df-base 13336  df-vtx 16169  df-clwwlk 16547  df-clwwlkn 16559

This theorem is referenced by: (None)