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Theorem clwwlknon1 27272
Description: The set of closed walks on vertex 𝑋 of length 1 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.)
Hypotheses
Ref Expression
clwwlknon1.v 𝑉 = (Vtx‘𝐺)
clwwlknon1.c 𝐶 = (ClWWalksNOn‘𝐺)
clwwlknon1.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
clwwlknon1 (𝑋𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
Distinct variable groups:   𝑤,𝐺   𝑤,𝑉   𝑤,𝑋
Allowed substitution hints:   𝐶(𝑤)   𝐸(𝑤)

Proof of Theorem clwwlknon1
StepHypRef Expression
1 clwwlknon1.c . . . 4 𝐶 = (ClWWalksNOn‘𝐺)
21oveqi 6806 . . 3 (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
32a1i 11 . 2 (𝑋𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4 clwwlknon 27262 . . 3 (𝑋(ClWWalksNOn‘𝐺)1) = {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
54a1i 11 . 2 (𝑋𝑉 → (𝑋(ClWWalksNOn‘𝐺)1) = {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
6 clwwlkn1 27197 . . . . 5 (𝑤 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
76anbi1i 610 . . . 4 ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
8 clwwlknon1.v . . . . . . . . . . . 12 𝑉 = (Vtx‘𝐺)
98eqcomi 2780 . . . . . . . . . . 11 (Vtx‘𝐺) = 𝑉
109wrdeqi 13524 . . . . . . . . . 10 Word (Vtx‘𝐺) = Word 𝑉
1110eleq2i 2842 . . . . . . . . 9 (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
1211biimpi 206 . . . . . . . 8 (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13123ad2ant2 1128 . . . . . . 7 (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → 𝑤 ∈ Word 𝑉)
1413ad2antrl 707 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 ∈ Word 𝑉)
1513adantr 466 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → 𝑤 ∈ Word 𝑉)
16 simpl1 1227 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (♯‘𝑤) = 1)
17 simpr 471 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
1815, 16, 173jca 1122 . . . . . . . 8 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋))
1918adantl 467 . . . . . . 7 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋))
20 wrdl1s1 13594 . . . . . . . 8 (𝑋𝑉 → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
2120adantr 466 . . . . . . 7 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
2219, 21mpbird 247 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 = ⟨“𝑋”⟩)
23 sneq 4326 . . . . . . . . . . . . 13 ((𝑤‘0) = 𝑋 → {(𝑤‘0)} = {𝑋})
24 clwwlknon1.e . . . . . . . . . . . . . . 15 𝐸 = (Edg‘𝐺)
2524eqcomi 2780 . . . . . . . . . . . . . 14 (Edg‘𝐺) = 𝐸
2625a1i 11 . . . . . . . . . . . . 13 ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
2723, 26eleq12d 2844 . . . . . . . . . . . 12 ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) ↔ {𝑋} ∈ 𝐸))
2827biimpd 219 . . . . . . . . . . 11 ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) → {𝑋} ∈ 𝐸))
2928a1i 11 . . . . . . . . . 10 (𝑋𝑉 → ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) → {𝑋} ∈ 𝐸)))
3029com13 88 . . . . . . . . 9 ({(𝑤‘0)} ∈ (Edg‘𝐺) → ((𝑤‘0) = 𝑋 → (𝑋𝑉 → {𝑋} ∈ 𝐸)))
31303ad2ant3 1129 . . . . . . . 8 (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → ((𝑤‘0) = 𝑋 → (𝑋𝑉 → {𝑋} ∈ 𝐸)))
3231imp 393 . . . . . . 7 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑋𝑉 → {𝑋} ∈ 𝐸))
3332impcom 394 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → {𝑋} ∈ 𝐸)
3414, 22, 33jca32 505 . . . . 5 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)))
35 fveq2 6332 . . . . . . . . . 10 (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36 s1len 13586 . . . . . . . . . 10 (♯‘⟨“𝑋”⟩) = 1
3735, 36syl6eq 2821 . . . . . . . . 9 (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = 1)
3837ad2antrl 707 . . . . . . . 8 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (♯‘𝑤) = 1)
3938adantl 467 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (♯‘𝑤) = 1)
408wrdeqi 13524 . . . . . . . . . 10 Word 𝑉 = Word (Vtx‘𝐺)
4140eleq2i 2842 . . . . . . . . 9 (𝑤 ∈ Word 𝑉𝑤 ∈ Word (Vtx‘𝐺))
4241biimpi 206 . . . . . . . 8 (𝑤 ∈ Word 𝑉𝑤 ∈ Word (Vtx‘𝐺))
4342ad2antrl 707 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → 𝑤 ∈ Word (Vtx‘𝐺))
44 fveq1 6331 . . . . . . . . . . . . . . 15 (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45 s1fv 13590 . . . . . . . . . . . . . . 15 (𝑋𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
4644, 45sylan9eq 2825 . . . . . . . . . . . . . 14 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → (𝑤‘0) = 𝑋)
4746eqcomd 2777 . . . . . . . . . . . . 13 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → 𝑋 = (𝑤‘0))
4847sneqd 4328 . . . . . . . . . . . 12 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → {𝑋} = {(𝑤‘0)})
4924a1i 11 . . . . . . . . . . . 12 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → 𝐸 = (Edg‘𝐺))
5048, 49eleq12d 2844 . . . . . . . . . . 11 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → ({𝑋} ∈ 𝐸 ↔ {(𝑤‘0)} ∈ (Edg‘𝐺)))
5150biimpd 219 . . . . . . . . . 10 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → ({𝑋} ∈ 𝐸 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5251impancom 439 . . . . . . . . 9 ((𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸) → (𝑋𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5352adantl 467 . . . . . . . 8 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5453impcom 394 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → {(𝑤‘0)} ∈ (Edg‘𝐺))
5539, 43, 543jca 1122 . . . . . 6 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
5646ex 397 . . . . . . . 8 (𝑤 = ⟨“𝑋”⟩ → (𝑋𝑉 → (𝑤‘0) = 𝑋))
5756ad2antrl 707 . . . . . . 7 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋𝑉 → (𝑤‘0) = 𝑋))
5857impcom 394 . . . . . 6 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (𝑤‘0) = 𝑋)
5955, 58jca 501 . . . . 5 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
6034, 59impbida 802 . . . 4 (𝑋𝑉 → ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
617, 60syl5bb 272 . . 3 (𝑋𝑉 → ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
6261rabbidva2 3336 . 2 (𝑋𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
633, 5, 623eqtrd 2809 1 (𝑋𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  {crab 3065  {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  ClWWalksNOncclwwlknon 27259
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  df-clwwlknon 27260
This theorem is referenced by:  clwwlknon1loop  27273  clwwlknon1nloop  27274
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