Theorem clwwlkn1loopb 16162

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 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))

Distinct variable groups:   𝑣,𝐺   𝑣,𝑊

Proof of Theorem clwwlkn1loopb

Step	Hyp	Ref	Expression
1		clwwlkn1 16160	. 2 ⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
2		wrdl1exs1 11177	. . . . . 6 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩)
3		fveq1 5628	. . . . . . . . . . . . . . 15 ⊢ (𝑊 = ⟨“𝑣”⟩ → (𝑊‘0) = (⟨“𝑣”⟩‘0))
4		s1fv 11174	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (⟨“𝑣”⟩‘0) = 𝑣)
5	3, 4	sylan9eq 2282	. . . . . . . . . . . . . 14 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊‘0) = 𝑣)
6	5	sneqd 3679	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → {(𝑊‘0)} = {𝑣})
7	6	eleq1d 2298	. . . . . . . . . . . 12 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {𝑣} ∈ (Edg‘𝐺)))
8	7	biimpd 144	. . . . . . . . . . 11 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → {𝑣} ∈ (Edg‘𝐺)))
9	8	ex 115	. . . . . . . . . 10 ⊢ (𝑊 = ⟨“𝑣”⟩ → (𝑣 ∈ (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → {𝑣} ∈ (Edg‘𝐺))))
10	9	com13 80	. . . . . . . . 9 ⊢ ({(𝑊‘0)} ∈ (Edg‘𝐺) → (𝑣 ∈ (Vtx‘𝐺) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺))))
11	10	imp 124	. . . . . . . 8 ⊢ (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺)))
12	11	ancld 325	. . . . . . 7 ⊢ (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
13	12	reximdva 2632	. . . . . 6 ⊢ ({(𝑊‘0)} ∈ (Edg‘𝐺) → (∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩ → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
14	2, 13	syl5com 29	. . . . 5 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
15	14	expcom 116	. . . 4 ⊢ ((♯‘𝑊) = 1 → (𝑊 ∈ Word (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))))
16	15	3imp 1217	. . 3 ⊢ (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
17		s1leng 11172	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (♯‘⟨“𝑣”⟩) = 1)
18	17	adantr 276	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → (♯‘⟨“𝑣”⟩) = 1)
19		s1cl 11169	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
20	19	adantr 276	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
21	4	eqcomd 2235	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (Vtx‘𝐺) → 𝑣 = (⟨“𝑣”⟩‘0))
22	21	sneqd 3679	. . . . . . . . . 10 ⊢ (𝑣 ∈ (Vtx‘𝐺) → {𝑣} = {(⟨“𝑣”⟩‘0)})
23	22	eleq1d 2298	. . . . . . . . 9 ⊢ (𝑣 ∈ (Vtx‘𝐺) → ({𝑣} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
24	23	biimpd 144	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → ({𝑣} ∈ (Edg‘𝐺) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
25	24	imp 124	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))
26	18, 20, 25	3jca 1201	. . . . . 6 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
27	26	adantrl 478	. . . . 5 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
28		fveqeq2 5638	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → ((♯‘𝑊) = 1 ↔ (♯‘⟨“𝑣”⟩) = 1))
29		eleq1 2292	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → (𝑊 ∈ Word (Vtx‘𝐺) ↔ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺)))
30	3	sneqd 3679	. . . . . . . 8 ⊢ (𝑊 = ⟨“𝑣”⟩ → {(𝑊‘0)} = {(⟨“𝑣”⟩‘0)})
31	30	eleq1d 2298	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
32	28, 29, 31	3anbi123d 1346	. . . . . 6 ⊢ (𝑊 = ⟨“𝑣”⟩ → (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))))
33	32	ad2antrl 490	. . . . 5 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))))
34	27, 33	mpbird 167	. . . 4 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
35	34	rexlimiva 2643	. . 3 ⊢ (∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
36	16, 35	impbii 126	. 2 ⊢ (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
37	1, 36	bitri 184	1 ⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∃wrex 2509  {csn 3666  ‘cfv 5318  (class class class)co 6007  0cc0 8010  1c1 8011  ♯chash 11009  Word cword 11084  ⟨“cs1 11163  Vtxcvtx 15829  Edgcedg 15874   ClWWalksN cclwwlkn 16146

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-er 6688  df-map 6805  df-en 6896  df-dom 6897  df-fin 6898  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-inn 9122  df-n0 9381  df-z 9458  df-uz 9734  df-fz 10217  df-fzo 10351  df-ihash 11010  df-word 11085  df-lsw 11130  df-s1 11164  df-ndx 13051  df-slot 13052  df-base 13054  df-vtx 15831  df-clwwlk 16135  df-clwwlkn 16147

This theorem is referenced by: (None)