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Theorem wksfval 28846
Description: The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)
Hypotheses
Ref Expression
wksfval.v 𝑉 = (Vtx‘𝐺)
wksfval.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
wksfval (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))))})
Distinct variable groups:   𝑓,𝐺,𝑘,𝑝   𝑓,𝐼,𝑝   𝑉,𝑝   𝑓,𝑊
Allowed substitution hints:   𝐼(𝑘)   𝑉(𝑓,𝑘)   𝑊(𝑘,𝑝)

Proof of Theorem wksfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-wlks 28836 . 2 Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))})
2 fveq2 6888 . . . . . . . 8 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3 wksfval.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
42, 3eqtr4di 2791 . . . . . . 7 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
54dmeqd 5903 . . . . . 6 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
6 wrdeq 14482 . . . . . 6 (dom (iEdg‘𝑔) = dom 𝐼 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
75, 6syl 17 . . . . 5 (𝑔 = 𝐺 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
87eleq2d 2820 . . . 4 (𝑔 = 𝐺 → (𝑓 ∈ Word dom (iEdg‘𝑔) ↔ 𝑓 ∈ Word dom 𝐼))
9 fveq2 6888 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10 wksfval.v . . . . . 6 𝑉 = (Vtx‘𝐺)
119, 10eqtr4di 2791 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1211feq3d 6701 . . . 4 (𝑔 = 𝐺 → (𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ↔ 𝑝:(0...(♯‘𝑓))⟶𝑉))
134fveq1d 6890 . . . . . . 7 (𝑔 = 𝐺 → ((iEdg‘𝑔)‘(𝑓𝑘)) = (𝐼‘(𝑓𝑘)))
1413eqeq1d 2735 . . . . . 6 (𝑔 = 𝐺 → (((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)} ↔ (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}))
1513sseq2d 4013 . . . . . 6 (𝑔 = 𝐺 → ({(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘)) ↔ {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))
1614, 15ifpbi23d 1081 . . . . 5 (𝑔 = 𝐺 → (if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))) ↔ if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)))))
1716ralbidv 3178 . . . 4 (𝑔 = 𝐺 → (∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))) ↔ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)))))
188, 12, 173anbi123d 1437 . . 3 (𝑔 = 𝐺 → ((𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(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))))))
1918opabbidv 5213 . 2 (𝑔 = 𝐺 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))} = {⟨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))))})
20 elex 3493 . 2 (𝐺𝑊𝐺 ∈ V)
21 3anass 1096 . . . 4 ((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 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))))
2221opabbii 5214 . . 3 {⟨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, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)))))}
233fvexi 6902 . . . . . 6 𝐼 ∈ V
2423dmex 7897 . . . . 5 dom 𝐼 ∈ V
25 wrdexg 14470 . . . . 5 (dom 𝐼 ∈ V → Word dom 𝐼 ∈ V)
2624, 25mp1i 13 . . . 4 (𝐺𝑊 → Word dom 𝐼 ∈ V)
27 ovex 7437 . . . . . 6 (0...(♯‘𝑓)) ∈ V
2810fvexi 6902 . . . . . . 7 𝑉 ∈ V
2928a1i 11 . . . . . 6 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → 𝑉 ∈ V)
30 mapex 8822 . . . . . 6 (((0...(♯‘𝑓)) ∈ V ∧ 𝑉 ∈ V) → {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
3127, 29, 30sylancr 588 . . . . 5 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
32 simpl 484 . . . . . . 7 ((𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)))) → 𝑝:(0...(♯‘𝑓))⟶𝑉)
3332ss2abi 4062 . . . . . 6 {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))} ⊆ {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉}
3433a1i 11 . . . . 5 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))} ⊆ {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉})
3531, 34ssexd 5323 . . . 4 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))} ∈ V)
3626, 35opabex3d 7947 . . 3 (𝐺𝑊 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘)))))} ∈ V)
3722, 36eqeltrid 2838 . 2 (𝐺𝑊 → {⟨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))))} ∈ V)
381, 19, 20, 37fvmptd3 7017 1 (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))))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  if-wif 1062  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wral 3062  Vcvv 3475  wss 3947  {csn 4627  {cpr 4629  {copab 5209  dom cdm 5675  wf 6536  cfv 6540  (class class class)co 7404  0cc0 11106  1c1 11107   + caddc 11109  ...cfz 13480  ..^cfzo 13623  chash 14286  Word cword 14460  Vtxcvtx 28236  iEdgciedg 28237  Walkscwlks 28833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-n0 12469  df-z 12555  df-uz 12819  df-fz 13481  df-fzo 13624  df-hash 14287  df-word 14461  df-wlks 28836
This theorem is referenced by:  iswlk  28847  wlkprop  28848  wlkv  28849
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