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Theorem wlks 25841
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 3184 . 2 (𝑉𝑋𝑉 ∈ V)
2 elex 3184 . 2 (𝐸𝑌𝐸 ∈ V)
3 df-wlk 25830 . . . 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 5233 . . . . . . . 8 (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
6 wrdeq 13131 . . . . . . . 8 (dom 𝑒 = dom 𝐸 → Word dom 𝑒 = Word dom 𝐸)
75, 6syl 17 . . . . . . 7 (𝑒 = 𝐸 → Word dom 𝑒 = Word dom 𝐸)
87ad2antll 760 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → Word dom 𝑒 = Word dom 𝐸)
98eleq2d 2672 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑓 ∈ Word dom 𝑒𝑓 ∈ Word dom 𝐸))
10 feq3 5927 . . . . . . 7 (𝑣 = 𝑉 → (𝑝:(0...(#‘𝑓))⟶𝑣𝑝:(0...(#‘𝑓))⟶𝑉))
1110adantr 479 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑝:(0...(#‘𝑓))⟶𝑣𝑝:(0...(#‘𝑓))⟶𝑉))
1211adantl 480 . . . . 5 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝑣 = 𝑉𝑒 = 𝐸)) → (𝑝:(0...(#‘𝑓))⟶𝑣𝑝:(0...(#‘𝑓))⟶𝑉))
13 fveq1 6087 . . . . . . . 8 (𝑒 = 𝐸 → (𝑒‘(𝑓𝑘)) = (𝐸‘(𝑓𝑘)))
1413eqeq1d 2611 . . . . . . 7 (𝑒 = 𝐸 → ((𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
1514ralbidv 2968 . . . . . 6 (𝑒 = 𝐸 → (∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
1615ad2antll 760 . . . . 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 1390 . . . 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 4642 . . 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 471 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝑉 ∈ V)
20 simpr 475 . . 3 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → 𝐸 ∈ V)
21 3anass 1034 . . . . 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 4643 . . . 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 6967 . . . . . . 7 (𝐸 ∈ V → dom 𝐸 ∈ V)
2423adantl 480 . . . . . 6 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → dom 𝐸 ∈ V)
25 wrdexg 13119 . . . . . 6 (dom 𝐸 ∈ V → Word dom 𝐸 ∈ V)
2624, 25syl 17 . . . . 5 ((𝑉 ∈ V ∧ 𝐸 ∈ V) → Word dom 𝐸 ∈ V)
27 fzfi 12591 . . . . . . 7 (0...(#‘𝑓)) ∈ Fin
2819adantr 479 . . . . . . 7 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑓 ∈ Word dom 𝐸) → 𝑉 ∈ V)
29 mapex 7728 . . . . . . 7 (((0...(#‘𝑓)) ∈ Fin ∧ 𝑉 ∈ V) → {𝑝𝑝:(0...(#‘𝑓))⟶𝑉} ∈ V)
3027, 28, 29sylancr 693 . . . . . 6 (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑓 ∈ Word dom 𝐸) → {𝑝𝑝:(0...(#‘𝑓))⟶𝑉} ∈ V)
31 simpl 471 . . . . . . . 8 ((𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) → 𝑝:(0...(#‘𝑓))⟶𝑉)
3231ss2abi 3636 . . . . . . 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 4728 . . . . 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 7015 . . . 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 2691 . . 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 6664 . 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 492 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2595  wral 2895  Vcvv 3172  wss 3539  {cpr 4126  {copab 4636  dom cdm 5028  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  Fincfn 7819  0cc0 9793  1c1 9794   + caddc 9796  ...cfz 12155  ..^cfzo 12292  #chash 12937  Word cword 13095   Walks cwalk 25820
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-card 8626  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-fzo 12293  df-hash 12938  df-word 13103  df-wlk 25830
This theorem is referenced by:  iswlk  25842  wlkcompim  25848  wlkelwrd  25852
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