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Theorem wlk2v2e 27135
Description: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that 𝐺 is a simple graph (without loops) only if 𝑋𝑌. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
Hypotheses
Ref Expression
wlk2v2e.i 𝐼 = ⟨“{𝑋, 𝑌}”⟩
wlk2v2e.f 𝐹 = ⟨“00”⟩
wlk2v2e.x 𝑋 ∈ V
wlk2v2e.y 𝑌 ∈ V
wlk2v2e.p 𝑃 = ⟨“𝑋𝑌𝑋”⟩
wlk2v2e.g 𝐺 = ⟨{𝑋, 𝑌}, 𝐼
Assertion
Ref Expression
wlk2v2e 𝐹(Walks‘𝐺)𝑃

Proof of Theorem wlk2v2e
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 wlk2v2e.g . . . . 5 𝐺 = ⟨{𝑋, 𝑌}, 𝐼
2 wlk2v2e.i . . . . . 6 𝐼 = ⟨“{𝑋, 𝑌}”⟩
32opeq2i 4437 . . . . 5 ⟨{𝑋, 𝑌}, 𝐼⟩ = ⟨{𝑋, 𝑌}, ⟨“{𝑋, 𝑌}”⟩⟩
41, 3eqtri 2673 . . . 4 𝐺 = ⟨{𝑋, 𝑌}, ⟨“{𝑋, 𝑌}”⟩⟩
5 wlk2v2e.x . . . . 5 𝑋 ∈ V
6 wlk2v2e.y . . . . 5 𝑌 ∈ V
7 uspgr2v1e2w 26188 . . . . 5 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → ⟨{𝑋, 𝑌}, ⟨“{𝑋, 𝑌}”⟩⟩ ∈ USPGraph)
85, 6, 7mp2an 708 . . . 4 ⟨{𝑋, 𝑌}, ⟨“{𝑋, 𝑌}”⟩⟩ ∈ USPGraph
94, 8eqeltri 2726 . . 3 𝐺 ∈ USPGraph
10 uspgrupgr 26116 . . 3 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
119, 10ax-mp 5 . 2 𝐺 ∈ UPGraph
12 wlk2v2e.f . . . . 5 𝐹 = ⟨“00”⟩
132, 12wlk2v2elem1 27133 . . . 4 𝐹 ∈ Word dom 𝐼
14 wlk2v2e.p . . . . . . . 8 𝑃 = ⟨“𝑋𝑌𝑋”⟩
155prid1 4329 . . . . . . . . 9 𝑋 ∈ {𝑋, 𝑌}
166prid2 4330 . . . . . . . . 9 𝑌 ∈ {𝑋, 𝑌}
17 s3cl 13670 . . . . . . . . 9 ((𝑋 ∈ {𝑋, 𝑌} ∧ 𝑌 ∈ {𝑋, 𝑌} ∧ 𝑋 ∈ {𝑋, 𝑌}) → ⟨“𝑋𝑌𝑋”⟩ ∈ Word {𝑋, 𝑌})
1815, 16, 15, 17mp3an 1464 . . . . . . . 8 ⟨“𝑋𝑌𝑋”⟩ ∈ Word {𝑋, 𝑌}
1914, 18eqeltri 2726 . . . . . . 7 𝑃 ∈ Word {𝑋, 𝑌}
20 wrdf 13342 . . . . . . 7 (𝑃 ∈ Word {𝑋, 𝑌} → 𝑃:(0..^(#‘𝑃))⟶{𝑋, 𝑌})
2119, 20ax-mp 5 . . . . . 6 𝑃:(0..^(#‘𝑃))⟶{𝑋, 𝑌}
2214fveq2i 6232 . . . . . . . . 9 (#‘𝑃) = (#‘⟨“𝑋𝑌𝑋”⟩)
23 s3len 13685 . . . . . . . . 9 (#‘⟨“𝑋𝑌𝑋”⟩) = 3
2422, 23eqtr2i 2674 . . . . . . . 8 3 = (#‘𝑃)
2524oveq2i 6701 . . . . . . 7 (0..^3) = (0..^(#‘𝑃))
2625feq2i 6075 . . . . . 6 (𝑃:(0..^3)⟶{𝑋, 𝑌} ↔ 𝑃:(0..^(#‘𝑃))⟶{𝑋, 𝑌})
2721, 26mpbir 221 . . . . 5 𝑃:(0..^3)⟶{𝑋, 𝑌}
2812fveq2i 6232 . . . . . . . . 9 (#‘𝐹) = (#‘⟨“00”⟩)
29 s2len 13680 . . . . . . . . 9 (#‘⟨“00”⟩) = 2
3028, 29eqtri 2673 . . . . . . . 8 (#‘𝐹) = 2
3130oveq2i 6701 . . . . . . 7 (0...(#‘𝐹)) = (0...2)
32 3z 11448 . . . . . . . . 9 3 ∈ ℤ
33 fzoval 12510 . . . . . . . . 9 (3 ∈ ℤ → (0..^3) = (0...(3 − 1)))
3432, 33ax-mp 5 . . . . . . . 8 (0..^3) = (0...(3 − 1))
35 3m1e2 11175 . . . . . . . . 9 (3 − 1) = 2
3635oveq2i 6701 . . . . . . . 8 (0...(3 − 1)) = (0...2)
3734, 36eqtr2i 2674 . . . . . . 7 (0...2) = (0..^3)
3831, 37eqtri 2673 . . . . . 6 (0...(#‘𝐹)) = (0..^3)
3938feq2i 6075 . . . . 5 (𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} ↔ 𝑃:(0..^3)⟶{𝑋, 𝑌})
4027, 39mpbir 221 . . . 4 𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌}
412, 12, 5, 6, 14wlk2v2elem2 27134 . . . 4 𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}
4213, 40, 413pm3.2i 1259 . . 3 (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})
431fveq2i 6232 . . . . 5 (Vtx‘𝐺) = (Vtx‘⟨{𝑋, 𝑌}, 𝐼⟩)
44 prex 4939 . . . . . 6 {𝑋, 𝑌} ∈ V
45 s1cli 13421 . . . . . . 7 ⟨“{𝑋, 𝑌}”⟩ ∈ Word V
462, 45eqeltri 2726 . . . . . 6 𝐼 ∈ Word V
47 opvtxfv 25929 . . . . . 6 (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (Vtx‘⟨{𝑋, 𝑌}, 𝐼⟩) = {𝑋, 𝑌})
4844, 46, 47mp2an 708 . . . . 5 (Vtx‘⟨{𝑋, 𝑌}, 𝐼⟩) = {𝑋, 𝑌}
4943, 48eqtr2i 2674 . . . 4 {𝑋, 𝑌} = (Vtx‘𝐺)
501fveq2i 6232 . . . . 5 (iEdg‘𝐺) = (iEdg‘⟨{𝑋, 𝑌}, 𝐼⟩)
51 opiedgfv 25932 . . . . . 6 (({𝑋, 𝑌} ∈ V ∧ 𝐼 ∈ Word V) → (iEdg‘⟨{𝑋, 𝑌}, 𝐼⟩) = 𝐼)
5244, 46, 51mp2an 708 . . . . 5 (iEdg‘⟨{𝑋, 𝑌}, 𝐼⟩) = 𝐼
5350, 52eqtr2i 2674 . . . 4 𝐼 = (iEdg‘𝐺)
5449, 53upgriswlk 26593 . . 3 (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶{𝑋, 𝑌} ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))
5542, 54mpbiri 248 . 2 (𝐺 ∈ UPGraph → 𝐹(Walks‘𝐺)𝑃)
5611, 55ax-mp 5 1 𝐹(Walks‘𝐺)𝑃
Colors of variables: wff setvar class
Syntax hints:  w3a 1054   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  {cpr 4212  cop 4216   class class class wbr 4685  dom cdm 5143  wf 5922  cfv 5926  (class class class)co 6690  0cc0 9974  1c1 9975   + caddc 9977  cmin 10304  2c2 11108  3c3 11109  cz 11415  ...cfz 12364  ..^cfzo 12504  #chash 13157  Word cword 13323  ⟨“cs1 13326  ⟨“cs2 13632  ⟨“cs3 13633  Vtxcvtx 25919  iEdgciedg 25920  UPGraphcupgr 26020  USPGraphcuspgr 26088  Walkscwlks 26548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1033  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-card 8803  df-cda 9028  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-xnn0 11402  df-z 11416  df-uz 11726  df-fz 12365  df-fzo 12505  df-hash 13158  df-word 13331  df-concat 13333  df-s1 13334  df-s2 13639  df-s3 13640  df-vtx 25921  df-iedg 25922  df-edg 25985  df-uhgr 25998  df-upgr 26022  df-uspgr 26090  df-wlks 26551
This theorem is referenced by: (None)
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