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Theorem iswlk 26797
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 4810 . . 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 26796 . . . . 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 1163 . . . 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 2830 . . 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 274 . 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 2832 . . . . . 6 (𝑓 = 𝐹 → (𝑓 ∈ Word dom 𝐼𝐹 ∈ Word dom 𝐼))
98adantr 472 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑓 ∈ Word dom 𝐼𝐹 ∈ Word dom 𝐼))
10 simpr 477 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → 𝑝 = 𝑃)
11 fveq2 6375 . . . . . . . 8 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
1211oveq2d 6858 . . . . . . 7 (𝑓 = 𝐹 → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
1312adantr 472 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
1410, 13feq12d 6211 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝:(0...(♯‘𝑓))⟶𝑉𝑃:(0...(♯‘𝐹))⟶𝑉))
1511oveq2d 6858 . . . . . . 7 (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
1615adantr 472 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
17 fveq1 6374 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑘) = (𝑃𝑘))
18 fveq1 6374 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
1917, 18eqeq12d 2780 . . . . . . . 8 (𝑝 = 𝑃 → ((𝑝𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
2019adantl 473 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝑝𝑘) = (𝑝‘(𝑘 + 1)) ↔ (𝑃𝑘) = (𝑃‘(𝑘 + 1))))
21 fveq1 6374 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
2221fveq2d 6379 . . . . . . . 8 (𝑓 = 𝐹 → (𝐼‘(𝑓𝑘)) = (𝐼‘(𝐹𝑘)))
2317sneqd 4346 . . . . . . . 8 (𝑝 = 𝑃 → {(𝑝𝑘)} = {(𝑃𝑘)})
2422, 23eqeqan12d 2781 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝐼‘(𝑓𝑘)) = {(𝑝𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
2517, 18preq12d 4431 . . . . . . . . 9 (𝑝 = 𝑃 → {(𝑝𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
2625adantl 473 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → {(𝑝𝑘), (𝑝‘(𝑘 + 1))} = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
2722adantr 472 . . . . . . . 8 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝐼‘(𝑓𝑘)) = (𝐼‘(𝐹𝑘)))
2826, 27sseq12d 3794 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ({(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)) ↔ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
2920, 24, 28ifpbi123d 1100 . . . . . 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 3300 . . . . 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 1560 . . . 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 5149 . . 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 1160 . 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 270 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 197  wa 384  if-wif 1085  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wss 3732  {csn 4334  {cpr 4336  cop 4340   class class class wbr 4809  {copab 4871  dom cdm 5277  wf 6064  cfv 6068  (class class class)co 6842  0cc0 10189  1c1 10190   + caddc 10192  ...cfz 12533  ..^cfzo 12673  chash 13321  Word cword 13486  Vtxcvtx 26165  iEdgciedg 26166  Walkscwlks 26783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-ifp 1086  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-er 7947  df-map 8062  df-pm 8063  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-fzo 12674  df-hash 13322  df-word 13487  df-wlks 26786
This theorem is referenced by:  wlkprop  26798  iswlkg  26800  wlkvtxeledg  26810  wlk1walk  26826  wlkres  26859  redwlk  26861  wlkp1  26870  wlkd  26875  lfgrwlkprop  26876  crctcshwlkn0  27006  upwlkwlk  42389  upgrwlkupwlk  42390
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