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Theorem clwwlkn1loopb 27199
 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
StepHypRef Expression
1 clwwlkn1 27197 . 2 (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
2 wrdl1exs1 13593 . . . . . 6 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩)
3 fveq1 6331 . . . . . . . . . . . . . . 15 (𝑊 = ⟨“𝑣”⟩ → (𝑊‘0) = (⟨“𝑣”⟩‘0))
4 s1fv 13590 . . . . . . . . . . . . . . 15 (𝑣 ∈ (Vtx‘𝐺) → (⟨“𝑣”⟩‘0) = 𝑣)
53, 4sylan9eq 2825 . . . . . . . . . . . . . 14 ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊‘0) = 𝑣)
65sneqd 4328 . . . . . . . . . . . . 13 ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → {(𝑊‘0)} = {𝑣})
76eleq1d 2835 . . . . . . . . . . . 12 ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {𝑣} ∈ (Edg‘𝐺)))
87biimpd 219 . . . . . . . . . . 11 ((𝑊 = ⟨“𝑣”⟩ ∧ 𝑣 ∈ (Vtx‘𝐺)) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → {𝑣} ∈ (Edg‘𝐺)))
98ex 397 . . . . . . . . . 10 (𝑊 = ⟨“𝑣”⟩ → (𝑣 ∈ (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → {𝑣} ∈ (Edg‘𝐺))))
109com13 88 . . . . . . . . 9 ({(𝑊‘0)} ∈ (Edg‘𝐺) → (𝑣 ∈ (Vtx‘𝐺) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺))))
1110imp 393 . . . . . . . 8 (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → {𝑣} ∈ (Edg‘𝐺)))
1211ancld 540 . . . . . . 7 (({(𝑊‘0)} ∈ (Edg‘𝐺) ∧ 𝑣 ∈ (Vtx‘𝐺)) → (𝑊 = ⟨“𝑣”⟩ → (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
1312reximdva 3165 . . . . . 6 ({(𝑊‘0)} ∈ (Edg‘𝐺) → (∃𝑣 ∈ (Vtx‘𝐺)𝑊 = ⟨“𝑣”⟩ → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
142, 13syl5com 31 . . . . 5 ((𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = 1) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))))
1514expcom 398 . . . 4 ((♯‘𝑊) = 1 → (𝑊 ∈ Word (Vtx‘𝐺) → ({(𝑊‘0)} ∈ (Edg‘𝐺) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))))
16153imp 1101 . . 3 (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) → ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
17 s1len 13586 . . . . . . . 8 (♯‘⟨“𝑣”⟩) = 1
1817a1i 11 . . . . . . 7 ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → (♯‘⟨“𝑣”⟩) = 1)
19 s1cl 13582 . . . . . . . 8 (𝑣 ∈ (Vtx‘𝐺) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
2019adantr 466 . . . . . . 7 ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺))
214eqcomd 2777 . . . . . . . . . . 11 (𝑣 ∈ (Vtx‘𝐺) → 𝑣 = (⟨“𝑣”⟩‘0))
2221sneqd 4328 . . . . . . . . . 10 (𝑣 ∈ (Vtx‘𝐺) → {𝑣} = {(⟨“𝑣”⟩‘0)})
2322eleq1d 2835 . . . . . . . . 9 (𝑣 ∈ (Vtx‘𝐺) → ({𝑣} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
2423biimpd 219 . . . . . . . 8 (𝑣 ∈ (Vtx‘𝐺) → ({𝑣} ∈ (Edg‘𝐺) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
2524imp 393 . . . . . . 7 ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))
2618, 20, 253jca 1122 . . . . . 6 ((𝑣 ∈ (Vtx‘𝐺) ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
2726adantrl 695 . . . . 5 ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
28 fveq2 6332 . . . . . . . 8 (𝑊 = ⟨“𝑣”⟩ → (♯‘𝑊) = (♯‘⟨“𝑣”⟩))
2928eqeq1d 2773 . . . . . . 7 (𝑊 = ⟨“𝑣”⟩ → ((♯‘𝑊) = 1 ↔ (♯‘⟨“𝑣”⟩) = 1))
30 eleq1 2838 . . . . . . 7 (𝑊 = ⟨“𝑣”⟩ → (𝑊 ∈ Word (Vtx‘𝐺) ↔ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺)))
313sneqd 4328 . . . . . . . 8 (𝑊 = ⟨“𝑣”⟩ → {(𝑊‘0)} = {(⟨“𝑣”⟩‘0)})
3231eleq1d 2835 . . . . . . 7 (𝑊 = ⟨“𝑣”⟩ → ({(𝑊‘0)} ∈ (Edg‘𝐺) ↔ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺)))
3329, 30, 323anbi123d 1547 . . . . . 6 (𝑊 = ⟨“𝑣”⟩ → (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ((♯‘⟨“𝑣”⟩) = 1 ∧ ⟨“𝑣”⟩ ∈ Word (Vtx‘𝐺) ∧ {(⟨“𝑣”⟩‘0)} ∈ (Edg‘𝐺))))
3433ad2antrl 707 . . . . 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))))
3527, 34mpbird 247 . . . 4 ((𝑣 ∈ (Vtx‘𝐺) ∧ (𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺))) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
3635rexlimiva 3176 . . 3 (∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)) → ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
3716, 36impbii 199 . 2 (((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
381, 37bitri 264 1 (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631   ∈ wcel 2145  ∃wrex 3062  {csn 4316  ‘cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139  ♯chash 13321  Word cword 13487  ⟨“cs1 13490  Vtxcvtx 26095  Edgcedg 26160   ClWWalksN cclwwlkn 27174 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-map 8011  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-card 8965  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-xnn0 11566  df-z 11580  df-uz 11889  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13495  df-lsw 13496  df-s1 13498  df-clwwlk 27132  df-clwwlkn 27176 This theorem is referenced by: (None)
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