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Theorem clWlkcompim 40982
Description: Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
Hypotheses
Ref Expression
isclWlke.v 𝑉 = (Vtx‘𝐺)
isclWlke.i 𝐼 = (iEdg‘𝐺)
clWlkcomp.1 𝐹 = (1st𝑊)
clWlkcomp.2 𝑃 = (2nd𝑊)
Assertion
Ref Expression
clWlkcompim (𝑊 ∈ (ClWalkS‘𝐺) → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
Distinct variable groups:   𝑘,𝐹   𝑘,𝐺   𝑃,𝑘
Allowed substitution hints:   𝐼(𝑘)   𝑉(𝑘)   𝑊(𝑘)

Proof of Theorem clWlkcompim
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6116 . . . 4 (𝑊 ∈ (ClWalkS‘𝐺) → 𝐺 ∈ V)
2 clwlkS 40973 . . . . . 6 (𝐺 ∈ V → (ClWalkS‘𝐺) = {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))})
32eleq2d 2672 . . . . 5 (𝐺 ∈ V → (𝑊 ∈ (ClWalkS‘𝐺) ↔ 𝑊 ∈ {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))}))
4 elopaelxp 5104 . . . . . . 7 (𝑊 ∈ {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))} → 𝑊 ∈ (V × V))
54anim2i 590 . . . . . 6 ((𝐺 ∈ V ∧ 𝑊 ∈ {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))}) → (𝐺 ∈ V ∧ 𝑊 ∈ (V × V)))
65ex 448 . . . . 5 (𝐺 ∈ V → (𝑊 ∈ {⟨𝑓, 𝑔⟩ ∣ (𝑓(1Walks‘𝐺)𝑔 ∧ (𝑔‘0) = (𝑔‘(#‘𝑓)))} → (𝐺 ∈ V ∧ 𝑊 ∈ (V × V))))
73, 6sylbid 228 . . . 4 (𝐺 ∈ V → (𝑊 ∈ (ClWalkS‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ (V × V))))
81, 7mpcom 37 . . 3 (𝑊 ∈ (ClWalkS‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ (V × V)))
9 isclWlke.v . . . 4 𝑉 = (Vtx‘𝐺)
10 isclWlke.i . . . 4 𝐼 = (iEdg‘𝐺)
11 clWlkcomp.1 . . . 4 𝐹 = (1st𝑊)
12 clWlkcomp.2 . . . 4 𝑃 = (2nd𝑊)
139, 10, 11, 12clWlkcomp 40981 . . 3 ((𝐺 ∈ V ∧ 𝑊 ∈ (V × V)) → (𝑊 ∈ (ClWalkS‘𝐺) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))))
148, 13syl 17 . 2 (𝑊 ∈ (ClWalkS‘𝐺) → (𝑊 ∈ (ClWalkS‘𝐺) ↔ ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))))
1514ibi 254 1 (𝑊 ∈ (ClWalkS‘𝐺) → ((𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  if-wif 1005   = wceq 1474  wcel 1976  wral 2895  Vcvv 3172  wss 3539  {csn 4124  {cpr 4126   class class class wbr 4577  {copab 4636   × cxp 5026  dom cdm 5028  wf 5786  cfv 5790  (class class class)co 6527  1st c1st 7034  2nd c2nd 7035  0cc0 9792  1c1 9793   + caddc 9795  ...cfz 12152  ..^cfzo 12289  #chash 12934  Word cword 13092  Vtxcvtx 40224  iEdgciedg 40225  1Walksc1wlks 40791  ClWalkScclwlks 40971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-card 8625  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10868  df-n0 11140  df-z 11211  df-uz 11520  df-fz 12153  df-fzo 12290  df-hash 12935  df-word 13100  df-1wlks 40795  df-clwlks 40972
This theorem is referenced by:  upgrclwlkcompim  40983  clwlksfclwwlk2wrd  41260  clwlksfclwwlk1hash  41262  clwlksf1clwwlklem0  41266
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