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Theorem iswlk 27308
Description: Properties of a pair of functions to be a walk. (Contributed by AV, 30-Dec-2020.)
Hypotheses
Ref Expression
wksfval.v 𝑉 = (Vtx‘𝐺)
wksfval.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
iswlk ((𝐺𝑊𝐹𝑈𝑃𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
Distinct variable groups:   𝑘,𝐺   𝑘,𝐹   𝑃,𝑘
Allowed substitution hints:   𝑈(𝑘)   𝐼(𝑘)   𝑉(𝑘)   𝑊(𝑘)   𝑍(𝑘)

Proof of Theorem iswlk
Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5063 . . 3 (𝐹(Walks‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺))
2 wksfval.v . . . . . 6 𝑉 = (Vtx‘𝐺)
3 wksfval.i . . . . . 6 𝐼 = (iEdg‘𝐺)
42, 3wksfval 27307 . . . . 5 (𝐺𝑊 → (Walks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))})
543ad2ant1 1127 . . . 4 ((𝐺𝑊𝐹𝑈𝑃𝑍) → (Walks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))})
65eleq2d 2902 . . 3 ((𝐺𝑊𝐹𝑈𝑃𝑍) → (⟨𝐹, 𝑃⟩ ∈ (Walks‘𝐺) ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))}))
71, 6syl5bb 284 . 2 ((𝐺𝑊𝐹𝑈𝑃𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ ⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))}))
8 eleq1 2904 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∈ Word dom 𝐼𝐹 ∈ Word dom 𝐼))
98adantr 481 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓 ∈ Word dom 𝐼𝐹 ∈ Word dom 𝐼))
10 simpr 485 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
11 fveq2 6666 . . . . . . . 8 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
1211oveq2d 7167 . . . . . . 7 (𝑓 = 𝐹 → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
1312adantr 481 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
1410, 13feq12d 6498 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝:(0...(♯‘𝑓))⟶𝑉𝑃:(0...(♯‘𝐹))⟶𝑉))
1511oveq2d 7167 . . . . . . 7 (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
1615adantr 481 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
17 fveq1 6665 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑘) = (𝑃𝑘))
18 fveq1 6665 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
1917, 18eqeq12d 2841 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
2019adantl 482 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
21 fveq1 6665 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
2221fveq2d 6670 . . . . . . . 8 (𝑓 = 𝐹 → (𝐼‘(𝑓𝑘)) = (𝐼‘(𝐹𝑘)))
2317sneqd 4575 . . . . . . . 8 (𝑝 = 𝑃 → {(𝑝𝑘)} = {(𝑃𝑘)})
2422, 23eqeqan12d 2842 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝐼‘(𝑓𝑘)) = {(𝑝𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
2517, 18preq12d 4675 . . . . . . . . 9 (𝑝 = 𝑃 → {(𝑝𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
2625adantl 482 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → {(𝑝𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
2722adantr 481 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝐼‘(𝑓𝑘)) = (𝐼‘(𝐹𝑘)))
2826, 27sseq12d 4003 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ({(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)) ↔ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
2920, 24, 28ifpbi123d 1071 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))) ↔ if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
3016, 29raleqbidv 3406 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))) ↔ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))
319, 14, 303anbi123d 1429 . . . 4 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)))) ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
3231opelopabga 5416 . . 3 ((𝐹𝑈𝑃𝑍) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))} ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
33323adant1 1124 . 2 ((𝐺𝑊𝐹𝑈𝑃𝑍) → (⟨𝐹, 𝑃⟩ ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))} ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
347, 33bitrd 280 1 ((𝐺𝑊𝐹𝑈𝑃𝑍) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  if-wif 1056  w3a 1081   = wceq 1530  wcel 2107  wral 3142  wss 3939  {csn 4563  {cpr 4565  cop 4569   class class class wbr 5062  {copab 5124  dom cdm 5553  wf 6347  cfv 6351  (class class class)co 7151  0cc0 10529  1c1 10530   + caddc 10532  ...cfz 12885  ..^cfzo 13026  chash 13683  Word cword 13854  Vtxcvtx 26697  iEdgciedg 26698  Walkscwlks 27294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-ifp 1057  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-nel 3128  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-int 4874  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-pred 6145  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7572  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-er 8282  df-map 8401  df-en 8502  df-dom 8503  df-sdom 8504  df-fin 8505  df-card 9360  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12886  df-fzo 13027  df-hash 13684  df-word 13855  df-wlks 27297
This theorem is referenced by:  wlkprop  27309  iswlkg  27311  wlkvtxeledg  27321  wlk1walk  27336  redwlk  27370  wlkp1  27379  wlkd  27384  lfgrwlkprop  27385  crctcshwlkn0  27515  upwlkwlk  43848  upgrwlkupwlk  43849
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