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Theorem iswlk 28558
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 5106 . . 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 28557 . . . . 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 1133 . . . 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 2823 . . 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 282 . 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 2825 . . . . . 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 6842 . . . . . . . 8 (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
1211oveq2d 7373 . . . . . . 7 (𝑓 = 𝐹 → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
1312adantr 481 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (0...(♯‘𝑓)) = (0...(♯‘𝐹)))
1410, 13feq12d 6656 . . . . 5 ((𝑓 = 𝐹𝑝 = 𝑃) → (𝑝:(0...(♯‘𝑓))⟶𝑉𝑃:(0...(♯‘𝐹))⟶𝑉))
1511oveq2d 7373 . . . . . . 7 (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
1615adantr 481 . . . . . 6 ((𝑓 = 𝐹𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
17 fveq1 6841 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝𝑘) = (𝑃𝑘))
18 fveq1 6841 . . . . . . . . 9 (𝑝 = 𝑃 → (𝑝‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)))
1917, 18eqeq12d 2752 . . . . . . . 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 6841 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓𝑘) = (𝐹𝑘))
2221fveq2d 6846 . . . . . . . 8 (𝑓 = 𝐹 → (𝐼‘(𝑓𝑘)) = (𝐼‘(𝐹𝑘)))
2317sneqd 4598 . . . . . . . 8 (𝑝 = 𝑃 → {(𝑝𝑘)} = {(𝑃𝑘)})
2422, 23eqeqan12d 2750 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ((𝐼‘(𝑓𝑘)) = {(𝑝𝑘)} ↔ (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}))
2517, 18preq12d 4702 . . . . . . . . 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 3977 . . . . . . 7 ((𝑓 = 𝐹𝑝 = 𝑃) → ({(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)) ↔ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))
2920, 24, 28ifpbi123d 1078 . . . . . 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 3319 . . . . 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 1436 . . . 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 5490 . . 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 1130 . 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 278 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 205  wa 396  if-wif 1061  w3a 1087   = wceq 1541  wcel 2106  wral 3064  wss 3910  {csn 4586  {cpr 4588  cop 4592   class class class wbr 5105  {copab 5167  dom cdm 5633  wf 6492  cfv 6496  (class class class)co 7357  0cc0 11051  1c1 11052   + caddc 11054  ...cfz 13424  ..^cfzo 13567  chash 14230  Word cword 14402  Vtxcvtx 27947  iEdgciedg 27948  Walkscwlks 28544
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-ifp 1062  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-fzo 13568  df-hash 14231  df-word 14403  df-wlks 28547
This theorem is referenced by:  wlkprop  28559  iswlkg  28561  wlkvtxeledg  28572  wlk1walk  28587  redwlk  28620  wlkp1  28629  wlkd  28634  lfgrwlkprop  28635  crctcshwlkn0  28766  upwlkwlk  46031  upgrwlkupwlk  46032
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