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Theorem disjxwwlkn 26789
 Description: Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.)
Hypotheses
Ref Expression
wwlksnextprop.x 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
wwlksnextprop.e 𝐸 = (Edg‘𝐺)
wwlksnextprop.y 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}
Assertion
Ref Expression
disjxwwlkn Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}
Distinct variable groups:   𝑤,𝐺   𝑤,𝑁   𝑤,𝑃   𝑦,𝐸   𝑥,𝑁,𝑦   𝑦,𝑃   𝑦,𝑋   𝑦,𝑌   𝑥,𝑤,𝐺   𝑦,𝑀   𝑥,𝑋
Allowed substitution hints:   𝑃(𝑥)   𝐸(𝑥,𝑤)   𝐺(𝑦)   𝑀(𝑥,𝑤)   𝑋(𝑤)   𝑌(𝑥,𝑤)

Proof of Theorem disjxwwlkn
StepHypRef Expression
1 simp1 1059 . . . . . 6 (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦)
21rgenw 2921 . . . . 5 𝑥𝑋 (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦)
3 ss2rab 3670 . . . . 5 ({𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} ⊆ {𝑥𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ↔ ∀𝑥𝑋 (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦))
42, 3mpbir 221 . . . 4 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} ⊆ {𝑥𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
5 wwlksnextprop.x . . . . . 6 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
6 wwlkssswwlksn 26732 . . . . . . 7 ((𝑁 + 1) WWalksN 𝐺) ⊆ (WWalks‘𝐺)
7 eqid 2620 . . . . . . . 8 (Vtx‘𝐺) = (Vtx‘𝐺)
87wwlkssswrd 26728 . . . . . . 7 (WWalks‘𝐺) ⊆ Word (Vtx‘𝐺)
96, 8sstri 3604 . . . . . 6 ((𝑁 + 1) WWalksN 𝐺) ⊆ Word (Vtx‘𝐺)
105, 9eqsstri 3627 . . . . 5 𝑋 ⊆ Word (Vtx‘𝐺)
11 rabss2 3677 . . . . 5 (𝑋 ⊆ Word (Vtx‘𝐺) → {𝑥𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦})
1210, 11ax-mp 5 . . . 4 {𝑥𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
134, 12sstri 3604 . . 3 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
1413rgenw 2921 . 2 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
15 disjxwrd 13437 . 2 Disj 𝑦𝑌 {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
16 disjss2 4614 . 2 (∀𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} → (Disj 𝑦𝑌 {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} → Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}))
1714, 15, 16mp2 9 1 Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1036   = wceq 1481   ∈ wcel 1988  ∀wral 2909  {crab 2913   ⊆ wss 3567  {cpr 4170  ⟨cop 4174  Disj wdisj 4611  ‘cfv 5876  (class class class)co 6635  0cc0 9921  1c1 9922   + caddc 9924  Word cword 13274   lastS clsw 13275   substr csubstr 13278  Vtxcvtx 25855  Edgcedg 25920  WWalkscwwlks 26698   WWalksN cwwlksn 26699 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-disj 4612  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-er 7727  df-map 7844  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-fzo 12450  df-hash 13101  df-word 13282  df-wwlks 26703  df-wwlksn 26704 This theorem is referenced by:  hashwwlksnext  26790
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