Theorem clwwlkn1loopb 29296

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 29294	. 2 ⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
2		wrdl1exs1 14563	. . . . . 6 ⊢ ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩)
3		fveq1 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑊 = ⟨“𝑣”⟩ → (𝑊‘0) = (⟨“𝑣”⟩‘0))
4		s1fv 14560	. . . . . . . . . . . . . . 15 ⊢ (𝑣 ∈ (Vtx‘𝐺) → (⟨“𝑣”⟩‘0) = 𝑣)
5	3, 4	sylan9eq 2793	. . . . . . . . . . . . . 14 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊‘0) = 𝑣)
6	5	sneqd 4641	. . . . . . . . . . . . 13 ⊢ ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → {(𝑊‘0)} = {𝑣})
7	6	eleq1d 2819	. . . . . . . . . . . 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 414	. . . . . . . . . 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 408	. . . . . . . 8 ⊢ (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺)))
12	11	ancld 552	. . . . . . 7 ⊢ (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
13	12	reximdva 3169	. . . . . 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 415	. . . 4 ⊢ ((♯‘𝑊) = 1 → (𝑊 ∈ Word (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))))
16	15	3imp 1112	. . 3 ⊢ (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
17		s1len 14556	. . . . . . . 8 ⊢ (♯‘⟨“𝑣”⟩) = 1
18	17	a1i 11	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → (♯‘⟨“𝑣”⟩) = 1)
19		s1cl 14552	. . . . . . . 8 ⊢ (𝑣 ∈ (Vtx‘𝐺) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
20	19	adantr 482	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
21	4	eqcomd 2739	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (Vtx‘𝐺) → 𝑣 = (⟨“𝑣”⟩‘0))
22	21	sneqd 4641	. . . . . . . . . 10 ⊢ (𝑣 ∈ (Vtx‘𝐺) → {𝑣} = {(⟨“𝑣”⟩‘0)})
23	22	eleq1d 2819	. . . . . . . . 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 408	. . . . . . 7 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))
26	18, 20, 25	3jca 1129	. . . . . 6 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
27	26	adantrl 715	. . . . 5 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
28		fveqeq2 6901	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → ((♯‘𝑊) = 1 ↔ (♯‘⟨“𝑣”⟩) = 1))
29		eleq1 2822	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → (𝑊 ∈ Word (Vtx‘𝐺) ↔ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺)))
30	3	sneqd 4641	. . . . . . . 8 ⊢ (𝑊 = ⟨“𝑣”⟩ → {(𝑊‘0)} = {(⟨“𝑣”⟩‘0)})
31	30	eleq1d 2819	. . . . . . 7 ⊢ (𝑊 = ⟨“𝑣”⟩ → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
32	28, 29, 31	3anbi123d 1437	. . . . . 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 727	. . . . 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 257	. . . 4 ⊢ ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
35	34	rexlimiva 3148	. . 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 275	1 ⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  {csn 4629  ‘cfv 6544  (class class class)co 7409  0cc0 11110  1c1 11111  ♯chash 14290  Word cword 14464  ⟨“cs1 14545  Vtxcvtx 28256  Edgcedg 28307   ClWWalksN cclwwlkn 29277

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9934  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-n0 12473  df-xnn0 12545  df-z 12559  df-uz 12823  df-fz 13485  df-fzo 13628  df-hash 14291  df-word 14465  df-lsw 14513  df-s1 14546  df-clwwlk 29235  df-clwwlkn 29278

This theorem is referenced by: (None)