Theorem clwwlkn1loopb 28407

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 28405	. 2 ⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
2		wrdl1exs1 14318	. . . . . 6 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩)
3		fveq1 6773	. . . . . . . . . . . . . . 15 ⊢ (𝑊 = ⟨“𝑣”⟩ → (𝑊‘0) = (⟨“𝑣”⟩‘0))
4		s1fv 14315	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (⟨“𝑣”⟩‘0) = 𝑣)
5	3, 4	sylan9eq 2798	. . . . . . . . . . . . . 14 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊‘0) = 𝑣)
6	5	sneqd 4573	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → {(𝑊‘0)} = {𝑣})
7	6	eleq1d 2823	. . . . . . . . . . . 12 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {𝑣} ∈ (Edg‘𝐺)))
8	7	biimpd 228	. . . . . . . . . . 11 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → {𝑣} ∈ (Edg‘𝐺)))
9	8	ex 413	. . . . . . . . . 10 ⊢ (𝑊 = ⟨“𝑣”⟩ → (𝑣 ∈ (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → {𝑣} ∈ (Edg‘𝐺))))
10	9	com13 88	. . . . . . . . 9 ⊢ ({(𝑊‘0)} ∈ (Edg‘𝐺) → (𝑣 ∈ (Vtx‘𝐺) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺))))
11	10	imp 407	. . . . . . . 8 ⊢ (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺)))
12	11	ancld 551	. . . . . . 7 ⊢ (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
13	12	reximdva 3203	. . . . . 6 ⊢ ({(𝑊‘0)} ∈ (Edg‘𝐺) → (∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩ → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
14	2, 13	syl5com 31	. . . . 5 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
15	14	expcom 414	. . . 4 ⊢ ((♯‘𝑊) = 1 → (𝑊 ∈ Word (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))))
16	15	3imp 1110	. . 3 ⊢ (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
17		s1len 14311	. . . . . . . 8 ⊢ (♯‘⟨“𝑣”⟩) = 1
18	17	a1i 11	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → (♯‘⟨“𝑣”⟩) = 1)
19		s1cl 14307	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
20	19	adantr 481	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
21	4	eqcomd 2744	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (Vtx‘𝐺) → 𝑣 = (⟨“𝑣”⟩‘0))
22	21	sneqd 4573	. . . . . . . . . 10 ⊢ (𝑣 ∈ (Vtx‘𝐺) → {𝑣} = {(⟨“𝑣”⟩‘0)})
23	22	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑣 ∈ (Vtx‘𝐺) → ({𝑣} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
24	23	biimpd 228	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → ({𝑣} ∈ (Edg‘𝐺) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
25	24	imp 407	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))
26	18, 20, 25	3jca 1127	. . . . . 6 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
27	26	adantrl 713	. . . . 5 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
28		fveqeq2 6783	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → ((♯‘𝑊) = 1 ↔ (♯‘⟨“𝑣”⟩) = 1))
29		eleq1 2826	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → (𝑊 ∈ Word (Vtx‘𝐺) ↔ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺)))
30	3	sneqd 4573	. . . . . . . 8 ⊢ (𝑊 = ⟨“𝑣”⟩ → {(𝑊‘0)} = {(⟨“𝑣”⟩‘0)})
31	30	eleq1d 2823	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
32	28, 29, 31	3anbi123d 1435	. . . . . 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 725	. . . . 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 256	. . . 4 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
35	34	rexlimiva 3210	. . 3 ⊢ (∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
36	16, 35	impbii 208	. 2 ⊢ (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
37	1, 36	bitri 274	1 ⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∃wrex 3065  {csn 4561  ‘cfv 6433  (class class class)co 7275  0cc0 10871  1c1 10872  ♯chash 14044  Word cword 14217  ⟨“cs1 14300  Vtxcvtx 27366  Edgcedg 27417   ClWWalksN cclwwlkn 28388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-hash 14045  df-word 14218  df-lsw 14266  df-s1 14301  df-clwwlk 28346  df-clwwlkn 28389

This theorem is referenced by: (None)