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Theorem clwwlknon1 27499
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 6935 . . 3 (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
32a1i 11 . 2 (𝑋𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4 clwwlknon 27492 . . 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 27431 . . . . 5 (𝑤 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
76anbi1i 617 . . . 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 2787 . . . . . . . . . . 11 (Vtx‘𝐺) = 𝑉
109wrdeqi 13625 . . . . . . . . . 10 Word (Vtx‘𝐺) = Word 𝑉
1110eleq2i 2851 . . . . . . . . 9 (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
1211biimpi 208 . . . . . . . 8 (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13123ad2ant2 1125 . . . . . . 7 (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → 𝑤 ∈ Word 𝑉)
1413ad2antrl 718 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 ∈ Word 𝑉)
1513adantr 474 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → 𝑤 ∈ Word 𝑉)
16 simpl1 1199 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (♯‘𝑤) = 1)
17 simpr 479 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
1815, 16, 173jca 1119 . . . . . . . 8 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋))
1918adantl 475 . . . . . . 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 13704 . . . . . . . 8 (𝑋𝑉 → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
2120adantr 474 . . . . . . 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 249 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 = ⟨“𝑋”⟩)
23 sneq 4408 . . . . . . . . . . . . 13 ((𝑤‘0) = 𝑋 → {(𝑤‘0)} = {𝑋})
24 clwwlknon1.e . . . . . . . . . . . . . . 15 𝐸 = (Edg‘𝐺)
2524eqcomi 2787 . . . . . . . . . . . . . 14 (Edg‘𝐺) = 𝐸
2625a1i 11 . . . . . . . . . . . . 13 ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
2723, 26eleq12d 2853 . . . . . . . . . . . 12 ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) ↔ {𝑋} ∈ 𝐸))
2827biimpd 221 . . . . . . . . . . 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 1126 . . . . . . . 8 (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → ((𝑤‘0) = 𝑋 → (𝑋𝑉 → {𝑋} ∈ 𝐸)))
3231imp 397 . . . . . . 7 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑋𝑉 → {𝑋} ∈ 𝐸))
3332impcom 398 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → {𝑋} ∈ 𝐸)
3414, 22, 33jca32 511 . . . . 5 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)))
35 fveq2 6446 . . . . . . . . . 10 (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36 s1len 13696 . . . . . . . . . 10 (♯‘⟨“𝑋”⟩) = 1
3735, 36syl6eq 2830 . . . . . . . . 9 (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = 1)
3837ad2antrl 718 . . . . . . . 8 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (♯‘𝑤) = 1)
3938adantl 475 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (♯‘𝑤) = 1)
408wrdeqi 13625 . . . . . . . . . 10 Word 𝑉 = Word (Vtx‘𝐺)
4140eleq2i 2851 . . . . . . . . 9 (𝑤 ∈ Word 𝑉𝑤 ∈ Word (Vtx‘𝐺))
4241biimpi 208 . . . . . . . 8 (𝑤 ∈ Word 𝑉𝑤 ∈ Word (Vtx‘𝐺))
4342ad2antrl 718 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → 𝑤 ∈ Word (Vtx‘𝐺))
44 fveq1 6445 . . . . . . . . . . . . . . 15 (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45 s1fv 13700 . . . . . . . . . . . . . . 15 (𝑋𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
4644, 45sylan9eq 2834 . . . . . . . . . . . . . 14 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → (𝑤‘0) = 𝑋)
4746eqcomd 2784 . . . . . . . . . . . . 13 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → 𝑋 = (𝑤‘0))
4847sneqd 4410 . . . . . . . . . . . 12 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → {𝑋} = {(𝑤‘0)})
4924a1i 11 . . . . . . . . . . . 12 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → 𝐸 = (Edg‘𝐺))
5048, 49eleq12d 2853 . . . . . . . . . . 11 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → ({𝑋} ∈ 𝐸 ↔ {(𝑤‘0)} ∈ (Edg‘𝐺)))
5150biimpd 221 . . . . . . . . . 10 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → ({𝑋} ∈ 𝐸 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5251impancom 445 . . . . . . . . 9 ((𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸) → (𝑋𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5352adantl 475 . . . . . . . 8 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5453impcom 398 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → {(𝑤‘0)} ∈ (Edg‘𝐺))
5539, 43, 543jca 1119 . . . . . 6 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
5646ex 403 . . . . . . . 8 (𝑤 = ⟨“𝑋”⟩ → (𝑋𝑉 → (𝑤‘0) = 𝑋))
5756ad2antrl 718 . . . . . . 7 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋𝑉 → (𝑤‘0) = 𝑋))
5857impcom 398 . . . . . 6 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (𝑤‘0) = 𝑋)
5955, 58jca 507 . . . . 5 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
6034, 59impbida 791 . . . 4 (𝑋𝑉 → ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
617, 60syl5bb 275 . . 3 (𝑋𝑉 → ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
6261rabbidva2 3383 . 2 (𝑋𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
633, 5, 623eqtrd 2818 1 (𝑋𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wcel 2107  {crab 3094  {csn 4398  cfv 6135  (class class class)co 6922  0cc0 10272  1c1 10273  chash 13435  Word cword 13599  ⟨“cs1 13685  Vtxcvtx 26344  Edgcedg 26395   ClWWalksN cclwwlkn 27413  ClWWalksNOncclwwlknon 27489
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-xnn0 11715  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-lsw 13653  df-s1 13686  df-clwwlk 27362  df-clwwlkn 27414  df-clwwlknon 27490
This theorem is referenced by:  clwwlknon1loop  27500  clwwlknon1nloop  27501
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