Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  clwwlknon1 Structured version   Visualization version   GIF version

Theorem clwwlknon1 27886
 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 7152 . . 3 (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1)
32a1i 11 . 2 (𝑋𝑉 → (𝑋𝐶1) = (𝑋(ClWWalksNOn‘𝐺)1))
4 clwwlknon 27879 . . 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 27830 . . . . 5 (𝑤 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
76anbi1i 626 . . . 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 2810 . . . . . . . . . . 11 (Vtx‘𝐺) = 𝑉
109wrdeqi 13884 . . . . . . . . . 10 Word (Vtx‘𝐺) = Word 𝑉
1110eleq2i 2884 . . . . . . . . 9 (𝑤 ∈ Word (Vtx‘𝐺) ↔ 𝑤 ∈ Word 𝑉)
1211biimpi 219 . . . . . . . 8 (𝑤 ∈ Word (Vtx‘𝐺) → 𝑤 ∈ Word 𝑉)
13123ad2ant2 1131 . . . . . . 7 (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → 𝑤 ∈ Word 𝑉)
1413ad2antrl 727 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 ∈ Word 𝑉)
1513adantr 484 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → 𝑤 ∈ Word 𝑉)
16 simpl1 1188 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (♯‘𝑤) = 1)
17 simpr 488 . . . . . . . . 9 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤‘0) = 𝑋)
1815, 16, 173jca 1125 . . . . . . . 8 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋))
1918adantl 485 . . . . . . 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 13963 . . . . . . . 8 (𝑋𝑉 → (𝑤 = ⟨“𝑋”⟩ ↔ (𝑤 ∈ Word 𝑉 ∧ (♯‘𝑤) = 1 ∧ (𝑤‘0) = 𝑋)))
2120adantr 484 . . . . . . 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 260 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → 𝑤 = ⟨“𝑋”⟩)
23 sneq 4538 . . . . . . . . . . . . 13 ((𝑤‘0) = 𝑋 → {(𝑤‘0)} = {𝑋})
24 clwwlknon1.e . . . . . . . . . . . . . . 15 𝐸 = (Edg‘𝐺)
2524eqcomi 2810 . . . . . . . . . . . . . 14 (Edg‘𝐺) = 𝐸
2625a1i 11 . . . . . . . . . . . . 13 ((𝑤‘0) = 𝑋 → (Edg‘𝐺) = 𝐸)
2723, 26eleq12d 2887 . . . . . . . . . . . 12 ((𝑤‘0) = 𝑋 → ({(𝑤‘0)} ∈ (Edg‘𝐺) ↔ {𝑋} ∈ 𝐸))
2827biimpd 232 . . . . . . . . . . 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 1132 . . . . . . . 8 (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) → ((𝑤‘0) = 𝑋 → (𝑋𝑉 → {𝑋} ∈ 𝐸)))
3231imp 410 . . . . . . 7 ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) → (𝑋𝑉 → {𝑋} ∈ 𝐸))
3332impcom 411 . . . . . 6 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → {𝑋} ∈ 𝐸)
3414, 22, 33jca32 519 . . . . 5 ((𝑋𝑉 ∧ (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋)) → (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)))
35 fveq2 6649 . . . . . . . . . 10 (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = (♯‘⟨“𝑋”⟩))
36 s1len 13955 . . . . . . . . . 10 (♯‘⟨“𝑋”⟩) = 1
3735, 36eqtrdi 2852 . . . . . . . . 9 (𝑤 = ⟨“𝑋”⟩ → (♯‘𝑤) = 1)
3837ad2antrl 727 . . . . . . . 8 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (♯‘𝑤) = 1)
3938adantl 485 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (♯‘𝑤) = 1)
408wrdeqi 13884 . . . . . . . . . 10 Word 𝑉 = Word (Vtx‘𝐺)
4140eleq2i 2884 . . . . . . . . 9 (𝑤 ∈ Word 𝑉𝑤 ∈ Word (Vtx‘𝐺))
4241biimpi 219 . . . . . . . 8 (𝑤 ∈ Word 𝑉𝑤 ∈ Word (Vtx‘𝐺))
4342ad2antrl 727 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → 𝑤 ∈ Word (Vtx‘𝐺))
44 fveq1 6648 . . . . . . . . . . . . . . 15 (𝑤 = ⟨“𝑋”⟩ → (𝑤‘0) = (⟨“𝑋”⟩‘0))
45 s1fv 13959 . . . . . . . . . . . . . . 15 (𝑋𝑉 → (⟨“𝑋”⟩‘0) = 𝑋)
4644, 45sylan9eq 2856 . . . . . . . . . . . . . 14 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → (𝑤‘0) = 𝑋)
4746eqcomd 2807 . . . . . . . . . . . . 13 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → 𝑋 = (𝑤‘0))
4847sneqd 4540 . . . . . . . . . . . 12 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → {𝑋} = {(𝑤‘0)})
4924a1i 11 . . . . . . . . . . . 12 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → 𝐸 = (Edg‘𝐺))
5048, 49eleq12d 2887 . . . . . . . . . . 11 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → ({𝑋} ∈ 𝐸 ↔ {(𝑤‘0)} ∈ (Edg‘𝐺)))
5150biimpd 232 . . . . . . . . . 10 ((𝑤 = ⟨“𝑋”⟩ ∧ 𝑋𝑉) → ({𝑋} ∈ 𝐸 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5251impancom 455 . . . . . . . . 9 ((𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸) → (𝑋𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5352adantl 485 . . . . . . . 8 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋𝑉 → {(𝑤‘0)} ∈ (Edg‘𝐺)))
5453impcom 411 . . . . . . 7 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → {(𝑤‘0)} ∈ (Edg‘𝐺))
5539, 43, 543jca 1125 . . . . . 6 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → ((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)))
5646ex 416 . . . . . . . 8 (𝑤 = ⟨“𝑋”⟩ → (𝑋𝑉 → (𝑤‘0) = 𝑋))
5756ad2antrl 727 . . . . . . 7 ((𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)) → (𝑋𝑉 → (𝑤‘0) = 𝑋))
5857impcom 411 . . . . . 6 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (𝑤‘0) = 𝑋)
5955, 58jca 515 . . . . 5 ((𝑋𝑉 ∧ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))) → (((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋))
6034, 59impbida 800 . . . 4 (𝑋𝑉 → ((((♯‘𝑤) = 1 ∧ 𝑤 ∈ Word (Vtx‘𝐺) ∧ {(𝑤‘0)} ∈ (Edg‘𝐺)) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
617, 60syl5bb 286 . . 3 (𝑋𝑉 → ((𝑤 ∈ (1 ClWWalksN 𝐺) ∧ (𝑤‘0) = 𝑋) ↔ (𝑤 ∈ Word 𝑉 ∧ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸))))
6261rabbidva2 3426 . 2 (𝑋𝑉 → {𝑤 ∈ (1 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
633, 5, 623eqtrd 2840 1 (𝑋𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  {crab 3113  {csn 4528  ‘cfv 6328  (class class class)co 7139  0cc0 10530  1c1 10531  ♯chash 13690  Word cword 13861  ⟨“cs1 13944  Vtxcvtx 26793  Edgcedg 26844   ClWWalksN cclwwlkn 27813  ClWWalksNOncclwwlknon 27876 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-card 9356  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-n0 11890  df-xnn0 11960  df-z 11974  df-uz 12236  df-fz 12890  df-fzo 13033  df-hash 13691  df-word 13862  df-lsw 13910  df-s1 13945  df-clwwlk 27771  df-clwwlkn 27814  df-clwwlknon 27877 This theorem is referenced by:  clwwlknon1loop  27887  clwwlknon1nloop  27888
 Copyright terms: Public domain W3C validator