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Theorem wlks 26047
 Description: The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.)
Assertion
Ref Expression
wlks ((𝑉𝑋𝐸𝑌) → (𝑉 Walks 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
Distinct variable groups:   𝑓,𝑘,𝑝,𝐸   𝑓,𝑉,𝑝
Allowed substitution hints:   𝑉(𝑘)   𝑋(𝑓,𝑘,𝑝)   𝑌(𝑓,𝑘,𝑝)

Proof of Theorem wlks
Dummy variables 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3185 . 2 (𝑉𝑋𝑉 ∈ V)
2 elex 3185 . 2 (𝐸𝑌𝐸 ∈ V)
3 df-wlk 26036 . . . 4 Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
43a1i 11 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})}))
5 dmeq 5246 . . . . . . . 8 (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
6 wrdeq 13182 . . . . . . . 8 (dom 𝑒 = dom 𝐸 → Word dom 𝑒 = Word dom 𝐸)
75, 6syl 17 . . . . . . 7 (𝑒 = 𝐸 → Word dom 𝑒 = Word dom 𝐸)
87ad2antll 761 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → Word dom 𝑒 = Word dom 𝐸)
98eleq2d 2673 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑓 ∈ Word dom 𝑒𝑓 ∈ Word dom 𝐸))
10 feq3 5941 . . . . . . 7 (𝑣 = 𝑉 → (𝑝:(0...(#‘𝑓))⟶𝑣𝑝:(0...(#‘𝑓))⟶𝑉))
1110adantr 480 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑝:(0...(#‘𝑓))⟶𝑣𝑝:(0...(#‘𝑓))⟶𝑉))
1211adantl 481 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑝:(0...(#‘𝑓))⟶𝑣𝑝:(0...(#‘𝑓))⟶𝑉))
13 fveq1 6102 . . . . . . . 8 (𝑒 = 𝐸 → (𝑒‘(𝑓𝑘)) = (𝐸‘(𝑓𝑘)))
1413eqeq1d 2612 . . . . . . 7 (𝑒 = 𝐸 → ((𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
1514ralbidv 2969 . . . . . 6 (𝑒 = 𝐸 → (∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
1615ad2antll 761 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
179, 12, 163anbi123d 1391 . . . 4 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → ((𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) ↔ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})))
1817opabbidv 4648 . . 3 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
19 simpl 472 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
20 simpr 476 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
21 3anass 1035 . . . . 5 ((𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) ↔ (𝑓 ∈ Word dom 𝐸 ∧ (𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})))
2221opabbii 4649 . . . 4 {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸 ∧ (𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))}
23 dmexg 6989 . . . . . . 7 (𝐸 ∈ V → dom 𝐸 ∈ V)
2423adantl 481 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → dom 𝐸 ∈ V)
25 wrdexg 13170 . . . . . 6 (dom 𝐸 ∈ V → Word dom 𝐸 ∈ V)
2624, 25syl 17 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Word dom 𝐸 ∈ V)
27 fzfi 12633 . . . . . . 7 (0...(#‘𝑓)) ∈ Fin
2819adantr 480 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑓 ∈ Word dom 𝐸) → 𝑉 ∈ V)
29 mapex 7750 . . . . . . 7 (((0...(#‘𝑓)) ∈ Fin ∧ 𝑉 ∈ V) → {𝑝𝑝:(0...(#‘𝑓))⟶𝑉} ∈ V)
3027, 28, 29sylancr 694 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑓 ∈ Word dom 𝐸) → {𝑝𝑝:(0...(#‘𝑓))⟶𝑉} ∈ V)
31 simpl 472 . . . . . . . 8 ((𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) → 𝑝:(0...(#‘𝑓))⟶𝑉)
3231ss2abi 3637 . . . . . . 7 {𝑝 ∣ (𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ⊆ {𝑝𝑝:(0...(#‘𝑓))⟶𝑉}
3332a1i 11 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑓 ∈ Word dom 𝐸) → {𝑝 ∣ (𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ⊆ {𝑝𝑝:(0...(#‘𝑓))⟶𝑉})
3430, 33ssexd 4733 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑓 ∈ Word dom 𝐸) → {𝑝 ∣ (𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
3526, 34opabex3d 7037 . . . 4 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸 ∧ (𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))} ∈ V)
3622, 35syl5eqel 2692 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
374, 18, 19, 20, 36ovmpt2d 6686 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 Walks 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
381, 2, 37syl2an 493 1 ((𝑉𝑋𝐸𝑌) → (𝑉 Walks 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  Vcvv 3173   ⊆ wss 3540  {cpr 4127  {copab 4642  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   Walks cwalk 26026 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-wlk 26036 This theorem is referenced by:  iswlk  26048  wlkcompim  26054  wlkelwrd  26058
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