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Theorem wlkelwrd 26058
 Description: The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
Assertion
Ref Expression
wlkelwrd (𝑊 ∈ (𝑉 Walks 𝐸) → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))

Proof of Theorem wlkelwrd
Dummy variables 𝑒 𝑓 𝑘 𝑝 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-wlk 26036 . . 3 Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})
2 vex 3176 . . . 4 𝑣 ∈ V
3 vex 3176 . . . 4 𝑒 ∈ V
4 wlks 26047 . . . . 5 ((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))})})
5 ovex 6577 . . . . 5 (𝑣 Walks 𝑒) ∈ V
64, 5syl6eqelr 2697 . . . 4 ((𝑣 ∈ V ∧ 𝑒 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ∈ V)
72, 3, 6mp2an 704 . . 3 {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} ∈ V
8 dmeq 5246 . . . . . . . 8 (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
9 wrdeq 13182 . . . . . . . 8 (dom 𝑒 = dom 𝐸 → Word dom 𝑒 = Word dom 𝐸)
108, 9syl 17 . . . . . . 7 (𝑒 = 𝐸 → Word dom 𝑒 = Word dom 𝐸)
1110eleq2d 2673 . . . . . 6 (𝑒 = 𝐸 → (𝑓 ∈ Word dom 𝑒𝑓 ∈ Word dom 𝐸))
1211adantl 481 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑓 ∈ Word dom 𝑒𝑓 ∈ Word dom 𝐸))
13 feq3 5941 . . . . . 6 (𝑣 = 𝑉 → (𝑝:(0...(#‘𝑓))⟶𝑣𝑝:(0...(#‘𝑓))⟶𝑉))
1413adantr 480 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑝:(0...(#‘𝑓))⟶𝑣𝑝:(0...(#‘𝑓))⟶𝑉))
15 fveq1 6102 . . . . . . . 8 (𝑒 = 𝐸 → (𝑒‘(𝑓𝑘)) = (𝐸‘(𝑓𝑘)))
1615adantl 481 . . . . . . 7 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑒‘(𝑓𝑘)) = (𝐸‘(𝑓𝑘)))
1716eqeq1d 2612 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → ((𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
1817ralbidv 2969 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → (∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
1912, 14, 183anbi123d 1391 . . . 4 ((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))})))
2019opabbidv 4648 . . 3 ((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))})})
211, 7, 20elovmpt2 6777 . 2 (𝑊 ∈ (𝑉 Walks 𝐸) ↔ (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})}))
22 eleq1 2676 . . . . . 6 (𝑓 = (1st𝑊) → (𝑓 ∈ Word dom 𝐸 ↔ (1st𝑊) ∈ Word dom 𝐸))
23 fveq2 6103 . . . . . . . 8 (𝑓 = (1st𝑊) → (#‘𝑓) = (#‘(1st𝑊)))
2423oveq2d 6565 . . . . . . 7 (𝑓 = (1st𝑊) → (0...(#‘𝑓)) = (0...(#‘(1st𝑊))))
2524feq2d 5944 . . . . . 6 (𝑓 = (1st𝑊) → (𝑝:(0...(#‘𝑓))⟶𝑉𝑝:(0...(#‘(1st𝑊)))⟶𝑉))
2623oveq2d 6565 . . . . . . 7 (𝑓 = (1st𝑊) → (0..^(#‘𝑓)) = (0..^(#‘(1st𝑊))))
27 fveq1 6102 . . . . . . . . 9 (𝑓 = (1st𝑊) → (𝑓𝑘) = ((1st𝑊)‘𝑘))
2827fveq2d 6107 . . . . . . . 8 (𝑓 = (1st𝑊) → (𝐸‘(𝑓𝑘)) = (𝐸‘((1st𝑊)‘𝑘)))
2928eqeq1d 2612 . . . . . . 7 (𝑓 = (1st𝑊) → ((𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘((1st𝑊)‘𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
3026, 29raleqbidv 3129 . . . . . 6 (𝑓 = (1st𝑊) → (∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}))
3122, 25, 303anbi123d 1391 . . . . 5 (𝑓 = (1st𝑊) → ((𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) ↔ ((1st𝑊) ∈ Word dom 𝐸𝑝:(0...(#‘(1st𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})))
32 feq1 5939 . . . . . 6 (𝑝 = (2nd𝑊) → (𝑝:(0...(#‘(1st𝑊)))⟶𝑉 ↔ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))
33 fveq1 6102 . . . . . . . . 9 (𝑝 = (2nd𝑊) → (𝑝𝑘) = ((2nd𝑊)‘𝑘))
34 fveq1 6102 . . . . . . . . 9 (𝑝 = (2nd𝑊) → (𝑝‘(𝑘 + 1)) = ((2nd𝑊)‘(𝑘 + 1)))
3533, 34preq12d 4220 . . . . . . . 8 (𝑝 = (2nd𝑊) → {(𝑝𝑘), (𝑝‘(𝑘 + 1))} = {((2nd𝑊)‘𝑘), ((2nd𝑊)‘(𝑘 + 1))})
3635eqeq2d 2620 . . . . . . 7 (𝑝 = (2nd𝑊) → ((𝐸‘((1st𝑊)‘𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ (𝐸‘((1st𝑊)‘𝑘)) = {((2nd𝑊)‘𝑘), ((2nd𝑊)‘(𝑘 + 1))}))
3736ralbidv 2969 . . . . . 6 (𝑝 = (2nd𝑊) → (∀𝑘 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ↔ ∀𝑘 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑘)) = {((2nd𝑊)‘𝑘), ((2nd𝑊)‘(𝑘 + 1))}))
3832, 373anbi23d 1394 . . . . 5 (𝑝 = (2nd𝑊) → (((1st𝑊) ∈ Word dom 𝐸𝑝:(0...(#‘(1st𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))}) ↔ ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑘)) = {((2nd𝑊)‘𝑘), ((2nd𝑊)‘(𝑘 + 1))})))
3931, 38elopabi 7120 . . . 4 (𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑘)) = {((2nd𝑊)‘𝑘), ((2nd𝑊)‘(𝑘 + 1))}))
40 3simpa 1051 . . . 4 (((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘(1st𝑊)))(𝐸‘((1st𝑊)‘𝑘)) = {((2nd𝑊)‘𝑘), ((2nd𝑊)‘(𝑘 + 1))}) → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))
4139, 40syl 17 . . 3 (𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})} → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))
42413ad2ant3 1077 . 2 ((𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑊 ∈ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})}) → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))
4321, 42sylbi 206 1 (𝑊 ∈ (𝑉 Walks 𝐸) → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173  {cpr 4127  {copab 4642  dom cdm 5038  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  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:  vfwlkniswwlkn  26234  2wlkeq  26235  usg2wlkeq  26236
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