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Theorem wwlksnextbij 41108
 Description: There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 18-Apr-2021.)
Hypotheses
Ref Expression
wwlksnextbij.v 𝑉 = (Vtx‘𝐺)
wwlksnextbij.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
wwlksnextbij (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸})
Distinct variable groups:   𝑓,𝐸,𝑛,𝑤   𝑓,𝐺,𝑤   𝑓,𝑁,𝑤   𝑓,𝑉,𝑛,𝑤   𝑓,𝑊,𝑛,𝑤
Allowed substitution hints:   𝐺(𝑛)   𝑁(𝑛)

Proof of Theorem wwlksnextbij
Dummy variables 𝑝 𝑡 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6577 . . . 4 ((𝑁 + 1) WWalkSN 𝐺) ∈ V
21a1i 11 . . 3 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ((𝑁 + 1) WWalkSN 𝐺) ∈ V)
3 rabexg 4739 . . 3 (((𝑁 + 1) WWalkSN 𝐺) ∈ V → {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ∈ V)
4 mptexg 6389 . . 3 ({𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ∈ V → (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)) ∈ V)
52, 3, 43syl 18 . 2 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)) ∈ V)
6 wwlksnextbij.v . . . 4 𝑉 = (Vtx‘𝐺)
7 wwlksnextbij.e . . . 4 𝐸 = (Edg‘𝐺)
8 eqid 2610 . . . 4 {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
9 preq2 4213 . . . . . 6 (𝑛 = 𝑝 → {( lastS ‘𝑊), 𝑛} = {( lastS ‘𝑊), 𝑝})
109eleq1d 2672 . . . . 5 (𝑛 = 𝑝 → ({( lastS ‘𝑊), 𝑛} ∈ 𝐸 ↔ {( lastS ‘𝑊), 𝑝} ∈ 𝐸))
1110cbvrabv 3172 . . . 4 {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸} = {𝑝𝑉 ∣ {( lastS ‘𝑊), 𝑝} ∈ 𝐸}
12 fveq2 6103 . . . . . . . 8 (𝑡 = 𝑤 → (#‘𝑡) = (#‘𝑤))
1312eqeq1d 2612 . . . . . . 7 (𝑡 = 𝑤 → ((#‘𝑡) = (𝑁 + 2) ↔ (#‘𝑤) = (𝑁 + 2)))
14 oveq1 6556 . . . . . . . 8 (𝑡 = 𝑤 → (𝑡 substr ⟨0, (𝑁 + 1)⟩) = (𝑤 substr ⟨0, (𝑁 + 1)⟩))
1514eqeq1d 2612 . . . . . . 7 (𝑡 = 𝑤 → ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ↔ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊))
16 fveq2 6103 . . . . . . . . 9 (𝑡 = 𝑤 → ( lastS ‘𝑡) = ( lastS ‘𝑤))
1716preq2d 4219 . . . . . . . 8 (𝑡 = 𝑤 → {( lastS ‘𝑊), ( lastS ‘𝑡)} = {( lastS ‘𝑊), ( lastS ‘𝑤)})
1817eleq1d 2672 . . . . . . 7 (𝑡 = 𝑤 → ({( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸 ↔ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸))
1913, 15, 183anbi123d 1391 . . . . . 6 (𝑡 = 𝑤 → (((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸) ↔ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)))
2019cbvrabv 3172 . . . . 5 {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}
2120mpteq1i 4667 . . . 4 (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)) = (𝑥 ∈ {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} ↦ ( lastS ‘𝑥))
226, 7, 8, 11, 21wwlksnextbij0 41107 . . 3 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸})
23 eqid 2610 . . . . . . 7 {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} = {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)}
246, 7, 23wwlksnextwrd 41103 . . . . . 6 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} = {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)})
2524eqcomd 2616 . . . . 5 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} = {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)})
2625mpteq1d 4666 . . . 4 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)) = (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)))
276, 7, 8wwlksnextwrd 41103 . . . . 5 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
2827eqcomd 2616 . . . 4 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → {𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)} = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})
29 eqidd 2611 . . . 4 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸} = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸})
3026, 28, 29f1oeq123d 6046 . . 3 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ((𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸} ↔ (𝑥 ∈ {𝑡 ∈ Word 𝑉 ∣ ((#‘𝑡) = (𝑁 + 2) ∧ (𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}))
3122, 30mpbird 246 . 2 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸})
32 f1oeq1 6040 . 2 (𝑓 = (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)) → (𝑓:{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸} ↔ (𝑥 ∈ {𝑡 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑡 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑡)} ∈ 𝐸)} ↦ ( lastS ‘𝑥)):{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}))
335, 31, 32elabd 3321 1 (𝑊 ∈ (𝑁 WWalkSN 𝐺) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑁 + 1) WWalkSN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {crab 2900  Vcvv 3173  {cpr 4127  ⟨cop 4131   ↦ cmpt 4643  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  #chash 12979  Word cword 13146   lastS clsw 13147   substr csubstr 13150  Vtxcvtx 25673  Edgcedga 25792   WWalkSN cwwlksn 41029 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-fal 1481  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-oadd 7451  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-2 10956  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-wwlks 41033  df-wwlksn 41034 This theorem is referenced by:  wwlksnexthasheq  41109
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