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Theorem 0wlk 26850
Description: A pair of an empty set (of edges) and a second set (of vertices) is a walk iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 30-Oct-2021.)
Hypothesis
Ref Expression
0wlk.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
0wlk (𝐺𝑈 → (∅(Walks‘𝐺)𝑃𝑃:(0...0)⟶𝑉))

Proof of Theorem 0wlk
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 0wlk.v . . 3 𝑉 = (Vtx‘𝐺)
2 eqid 2621 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
31, 2iswlkg 26386 . 2 (𝐺𝑈 → (∅(Walks‘𝐺)𝑃 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘∅))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))))))
4 ral0 4050 . . . . 5 𝑘 ∈ ∅ if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))
5 hash0 13101 . . . . . . . 8 (#‘∅) = 0
65oveq2i 6618 . . . . . . 7 (0..^(#‘∅)) = (0..^0)
7 fzo0 12436 . . . . . . 7 (0..^0) = ∅
86, 7eqtri 2643 . . . . . 6 (0..^(#‘∅)) = ∅
98raleqi 3131 . . . . 5 (∀𝑘 ∈ (0..^(#‘∅))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))) ↔ ∀𝑘 ∈ ∅ if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘))))
104, 9mpbir 221 . . . 4 𝑘 ∈ (0..^(#‘∅))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))
1110biantru 526 . . 3 ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉) ↔ ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(#‘∅))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))))
125eqcomi 2630 . . . . . 6 0 = (#‘∅)
1312oveq2i 6618 . . . . 5 (0...0) = (0...(#‘∅))
1413feq2i 5996 . . . 4 (𝑃:(0...0)⟶𝑉𝑃:(0...(#‘∅))⟶𝑉)
15 wrd0 13272 . . . . 5 ∅ ∈ Word dom (iEdg‘𝐺)
1615biantrur 527 . . . 4 (𝑃:(0...(#‘∅))⟶𝑉 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉))
1714, 16bitri 264 . . 3 (𝑃:(0...0)⟶𝑉 ↔ (∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉))
18 df-3an 1038 . . 3 ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘∅))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))) ↔ ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(#‘∅))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))))
1911, 17, 183bitr4ri 293 . 2 ((∅ ∈ Word dom (iEdg‘𝐺) ∧ 𝑃:(0...(#‘∅))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘∅))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), ((iEdg‘𝐺)‘(∅‘𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ ((iEdg‘𝐺)‘(∅‘𝑘)))) ↔ 𝑃:(0...0)⟶𝑉)
203, 19syl6bb 276 1 (𝐺𝑈 → (∅(Walks‘𝐺)𝑃𝑃:(0...0)⟶𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  if-wif 1011  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wss 3556  c0 3893  {csn 4150  {cpr 4152   class class class wbr 4615  dom cdm 5076  wf 5845  cfv 5849  (class class class)co 6607  0cc0 9883  1c1 9884   + caddc 9886  ...cfz 12271  ..^cfzo 12409  #chash 13060  Word cword 13233  Vtxcvtx 25781  iEdgciedg 25782  Walkscwlks 26369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1012  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-er 7690  df-map 7807  df-pm 7808  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-card 8712  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-n0 11240  df-z 11325  df-uz 11635  df-fz 12272  df-fzo 12410  df-hash 13061  df-word 13241  df-wlks 26372
This theorem is referenced by:  is0wlk  26851  0wlkon  26854  0trl  26856  0clwlk  26864
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