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Theorem iswlk 29628
Description: Properties of a pair of functions to be/represent 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 5144 . . 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 29627 . . . . 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 1134 . . . 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 2827 . . 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, 6bitrid 283 . 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 2829 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∈ Word dom 𝐼𝐹 ∈ Word dom 𝐼))
98adantr 480 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓 ∈ Word dom 𝐼𝐹 ∈ Word dom 𝐼))
10 simpr 484 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
11 fveq2 6906 . . . . . . . 8 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
1211oveq2d 7447 . . . . . . 7 (𝑓 = 𝐹 → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
1312adantr 480 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
1410, 13feq12d 6724 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝:(0...(♯‘𝑓))⟶𝑉𝑃:(0...(♯‘𝐹))⟶𝑉))
1511oveq2d 7447 . . . . . . 7 (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
1615adantr 480 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
17 fveq1 6905 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑘) = (𝑃𝑘))
18 fveq1 6905 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
1917, 18eqeq12d 2753 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
2019adantl 481 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
21 fveq1 6905 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
2221fveq2d 6910 . . . . . . . 8 (𝑓 = 𝐹 → (𝐼‘(𝑓𝑘)) = (𝐼‘(𝐹𝑘)))
2317sneqd 4638 . . . . . . . 8 (𝑝 = 𝑃 → {(𝑝𝑘)} = {(𝑃𝑘)})
2422, 23eqeqan12d 2751 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝐼‘(𝑓𝑘)) = {(𝑝𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
2517, 18preq12d 4741 . . . . . . . . 9 (𝑝 = 𝑃 → {(𝑝𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
2625adantl 481 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → {(𝑝𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
2722adantr 480 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝐼‘(𝑓𝑘)) = (𝐼‘(𝐹𝑘)))
2826, 27sseq12d 4017 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ({(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)) ↔ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
2920, 24, 28ifpbi123d 1079 . . . . . 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 3346 . . . . 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 1438 . . . 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 5538 . . 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 1131 . 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 279 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 206  wa 395  if-wif 1063  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951  {csn 4626  {cpr 4628  cop 4632   class class class wbr 5143  {copab 5205  dom cdm 5685  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158  ...cfz 13547  ..^cfzo 13694  chash 14369  Word cword 14552  Vtxcvtx 29013  iEdgciedg 29014  Walkscwlks 29614
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ifp 1064  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-n0 12527  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-hash 14370  df-word 14553  df-wlks 29617
This theorem is referenced by:  wlkprop  29629  iswlkg  29631  wlkvtxeledg  29642  wlk1walk  29657  redwlk  29690  wlkp1  29699  wlkd  29704  lfgrwlkprop  29705  crctcshwlkn0  29841  upwlkwlk  48055  upgrwlkupwlk  48056
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