Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  upwlksfval Structured version   Visualization version   GIF version

Theorem upwlksfval 47979
Description: The set of simple walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.)
Hypotheses
Ref Expression
upwlksfval.v 𝑉 = (Vtx‘𝐺)
upwlksfval.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
upwlksfval (𝐺𝑊 → (UPWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
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 upwlksfval
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 df-upwlks 47978 . 2 UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
2 fveq2 6907 . . . . . . . 8 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
3 upwlksfval.i . . . . . . . 8 𝐼 = (iEdg‘𝐺)
42, 3eqtr4di 2793 . . . . . . 7 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
54dmeqd 5919 . . . . . 6 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
6 wrdeq 14571 . . . . . 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 2825 . . . 4 (𝑔 = 𝐺 → (𝑓 ∈ Word dom (iEdg‘𝑔) ↔ 𝑓 ∈ Word dom 𝐼))
9 fveq2 6907 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
10 upwlksfval.v . . . . . 6 𝑉 = (Vtx‘𝐺)
119, 10eqtr4di 2793 . . . . 5 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1211feq3d 6724 . . . 4 (𝑔 = 𝐺 → (𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ↔ 𝑝:(0...(♯‘𝑓))⟶𝑉))
134fveq1d 6909 . . . . . 6 (𝑔 = 𝐺 → ((iEdg‘𝑔)‘(𝑓𝑘)) = (𝐼‘(𝑓𝑘)))
1413eqeq1d 2737 . . . . 5 (𝑔 = 𝐺 → (((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
1514ralbidv 3176 . . . 4 (𝑔 = 𝐺 → (∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
168, 12, 153anbi123d 1435 . . 3 (𝑔 = 𝐺 → ((𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) ↔ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})))
1716opabbidv 5214 . 2 (𝑔 = 𝐺 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
18 elex 3499 . 2 (𝐺𝑊𝐺 ∈ V)
19 3anass 1094 . . . 4 ((𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) ↔ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})))
2019opabbii 5215 . . 3 {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))}
213fvexi 6921 . . . . . 6 𝐼 ∈ V
2221dmex 7932 . . . . 5 dom 𝐼 ∈ V
23 wrdexg 14559 . . . . 5 (dom 𝐼 ∈ V → Word dom 𝐼 ∈ V)
2422, 23mp1i 13 . . . 4 (𝐺𝑊 → Word dom 𝐼 ∈ V)
25 ovex 7464 . . . . . 6 (0...(♯‘𝑓)) ∈ V
2610fvexi 6921 . . . . . . 7 𝑉 ∈ V
2726a1i 11 . . . . . 6 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → 𝑉 ∈ V)
28 mapex 7962 . . . . . 6 (((0...(♯‘𝑓)) ∈ V ∧ 𝑉 ∈ V) → {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
2925, 27, 28sylancr 587 . . . . 5 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
30 simpl 482 . . . . . . 7 ((𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) → 𝑝:(0...(♯‘𝑓))⟶𝑉)
3130ss2abi 4077 . . . . . 6 {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ⊆ {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉}
3231a1i 11 . . . . 5 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ⊆ {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉})
3329, 32ssexd 5330 . . . 4 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
3424, 33opabex3d 7989 . . 3 (𝐺𝑊 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))} ∈ V)
3520, 34eqeltrid 2843 . 2 (𝐺𝑊 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
361, 17, 18, 35fvmptd3 7039 1 (𝐺𝑊 → (UPWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  Vcvv 3478  wss 3963  {cpr 4633  {copab 5210  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   + caddc 11156  ...cfz 13544  ..^cfzo 13691  chash 14366  Word cword 14549  Vtxcvtx 29028  iEdgciedg 29029  UPWalkscupwlks 47977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-fzo 13692  df-hash 14367  df-word 14550  df-upwlks 47978
This theorem is referenced by:  isupwlk  47980  isupwlkg  47981  upwlkbprop  47982
  Copyright terms: Public domain W3C validator