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 42268
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 42267 . . 3 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))})})
21a1i 11 . 2 (𝐺𝑊 → 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))})}))
3 fveq2 6348 . . . . . . . . 9 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
4 upwlksfval.i . . . . . . . . 9 𝐼 = (iEdg‘𝐺)
53, 4syl6eqr 2826 . . . . . . . 8 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
65dmeqd 5476 . . . . . . 7 (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
7 wrdeq 13545 . . . . . . 7 (dom (iEdg‘𝑔) = dom 𝐼 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
86, 7syl 17 . . . . . 6 (𝑔 = 𝐺 → Word dom (iEdg‘𝑔) = Word dom 𝐼)
98eleq2d 2839 . . . . 5 (𝑔 = 𝐺 → (𝑓 ∈ Word dom (iEdg‘𝑔) ↔ 𝑓 ∈ Word dom 𝐼))
10 fveq2 6348 . . . . . . 7 (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
11 upwlksfval.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
1210, 11syl6eqr 2826 . . . . . 6 (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
1312feq3d 6183 . . . . 5 (𝑔 = 𝐺 → (𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ↔ 𝑝:(0...(♯‘𝑓))⟶𝑉))
145fveq1d 6350 . . . . . . 7 (𝑔 = 𝐺 → ((iEdg‘𝑔)‘(𝑓𝑘)) = (𝐼‘(𝑓𝑘)))
1514eqeq1d 2776 . . . . . 6 (𝑔 = 𝐺 → (((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
1615ralbidv 3138 . . . . 5 (𝑔 = 𝐺 → (∀𝑘 ∈ (0..^(♯‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
179, 13, 163anbi123d 1550 . . . 4 (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))})))
1817opabbidv 4863 . . 3 (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))})})
1918adantl 468 . 2 ((𝐺𝑊𝑔 = 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))})})
20 elex 3369 . 2 (𝐺𝑊𝐺 ∈ V)
21 3anass 1107 . . . 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))})))
2221opabbii 4864 . . 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))}))}
234fvexi 6360 . . . . . 6 𝐼 ∈ V
2423dmex 7267 . . . . 5 dom 𝐼 ∈ V
25 wrdexg 13533 . . . . 5 (dom 𝐼 ∈ V → Word dom 𝐼 ∈ V)
2624, 25mp1i 13 . . . 4 (𝐺𝑊 → Word dom 𝐼 ∈ V)
27 ovex 6844 . . . . . 6 (0...(♯‘𝑓)) ∈ V
2811fvexi 6360 . . . . . . 7 𝑉 ∈ V
2928a1i 11 . . . . . 6 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → 𝑉 ∈ V)
30 mapex 8036 . . . . . 6 (((0...(♯‘𝑓)) ∈ V ∧ 𝑉 ∈ V) → {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
3127, 29, 30sylancr 576 . . . . 5 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉} ∈ V)
32 simpl 469 . . . . . . 7 ((𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) → 𝑝:(0...(♯‘𝑓))⟶𝑉)
3332ss2abi 3830 . . . . . 6 {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ⊆ {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉}
3433a1i 11 . . . . 5 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ⊆ {𝑝𝑝:(0...(♯‘𝑓))⟶𝑉})
3531, 34ssexd 4953 . . . 4 ((𝐺𝑊𝑓 ∈ Word dom 𝐼) → {𝑝 ∣ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
3626, 35opabex3d 7313 . . 3 (𝐺𝑊 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼 ∧ (𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))} ∈ V)
3722, 36syl5eqel 2857 . 2 (𝐺𝑊 → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(♯‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
382, 19, 20, 37fvmptd 6447 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 383  w3a 1098   = wceq 1634  wcel 2148  {cab 2760  wral 3064  Vcvv 3355  wss 3729  {cpr 4328  {copab 4859  cmpt 4876  dom cdm 5263  wf 6038  cfv 6042  (class class class)co 6812  0cc0 10159  1c1 10160   + caddc 10162  ...cfz 12555  ..^cfzo 12695  chash 13343  Word cword 13509  Vtxcvtx 26116  iEdgciedg 26117  UPWalkscupwlks 42266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-8 2150  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-rep 4917  ax-sep 4928  ax-nul 4936  ax-pow 4988  ax-pr 5048  ax-un 7117  ax-cnex 10215  ax-resscn 10216  ax-1cn 10217  ax-icn 10218  ax-addcl 10219  ax-addrcl 10220  ax-mulcl 10221  ax-mulrcl 10222  ax-mulcom 10223  ax-addass 10224  ax-mulass 10225  ax-distr 10226  ax-i2m1 10227  ax-1ne0 10228  ax-1rid 10229  ax-rnegex 10230  ax-rrecex 10231  ax-cnre 10232  ax-pre-lttri 10233  ax-pre-lttrn 10234  ax-pre-ltadd 10235  ax-pre-mulgt0 10236
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3or 1099  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ne 2947  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3071  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-pss 3745  df-nul 4074  df-if 4236  df-pw 4309  df-sn 4327  df-pr 4329  df-tp 4331  df-op 4333  df-uni 4586  df-int 4623  df-iun 4667  df-br 4798  df-opab 4860  df-mpt 4877  df-tr 4900  df-id 5171  df-eprel 5176  df-po 5184  df-so 5185  df-fr 5222  df-we 5224  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-rn 5274  df-res 5275  df-ima 5276  df-pred 5834  df-ord 5880  df-on 5881  df-lim 5882  df-suc 5883  df-iota 6005  df-fun 6044  df-fn 6045  df-f 6046  df-f1 6047  df-fo 6048  df-f1o 6049  df-fv 6050  df-riota 6773  df-ov 6815  df-oprab 6816  df-mpt2 6817  df-om 7234  df-1st 7336  df-2nd 7337  df-wrecs 7580  df-recs 7642  df-rdg 7680  df-1o 7734  df-er 7917  df-map 8032  df-pm 8033  df-en 8131  df-dom 8132  df-sdom 8133  df-fin 8134  df-card 8986  df-pnf 10299  df-mnf 10300  df-xr 10301  df-ltxr 10302  df-le 10303  df-sub 10491  df-neg 10492  df-nn 11244  df-n0 11517  df-z 11602  df-uz 11911  df-fz 12556  df-fzo 12696  df-hash 13344  df-word 13517  df-upwlks 42267
This theorem is referenced by:  isupwlk  42269  isupwlkg  42270  upwlkbprop  42271
  Copyright terms: Public domain W3C validator