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Theorem disjxwwlkn 26273
 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.)
Hypotheses
Ref Expression
wwlkextprop.x 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))
wwlkextprop.y 𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}
Assertion
Ref Expression
disjxwwlkn Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}
Distinct variable groups:   𝑤,𝐸   𝑤,𝑁   𝑤,𝑃   𝑤,𝑉   𝑦,𝐸   𝑥,𝑁,𝑦,𝑤   𝑦,𝑃   𝑦,𝑋   𝑦,𝑌   𝑥,𝑀,𝑦   𝑥,𝑉,𝑦   𝑥,𝑋   𝑥,𝑌
Allowed substitution hints:   𝑃(𝑥)   𝐸(𝑥)   𝑀(𝑤)   𝑋(𝑤)   𝑌(𝑤)

Proof of Theorem disjxwwlkn
StepHypRef Expression
1 simp1 1054 . . . . . 6 (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦)
21rgenw 2908 . . . . 5 𝑥𝑋 (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦)
3 ss2rab 3641 . . . . 5 ({𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)} ⊆ {𝑥𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ↔ ∀𝑥𝑋 (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦))
42, 3mpbir 220 . . . 4 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)} ⊆ {𝑥𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
5 wwlkextprop.x . . . . . 6 𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))
6 wwlksswwlkn 26231 . . . . . . 7 ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ⊆ (𝑉 WWalks 𝐸)
7 wwlksswrd 26216 . . . . . . 7 (𝑉 WWalks 𝐸) ⊆ Word 𝑉
86, 7sstri 3577 . . . . . 6 ((𝑉 WWalksN 𝐸)‘(𝑁 + 1)) ⊆ Word 𝑉
95, 8eqsstri 3598 . . . . 5 𝑋 ⊆ Word 𝑉
10 rabss2 3648 . . . . 5 (𝑋 ⊆ Word 𝑉 → {𝑥𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦})
119, 10ax-mp 5 . . . 4 {𝑥𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
124, 11sstri 3577 . . 3 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
1312rgenw 2908 . 2 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
14 disjxwrd 13307 . 2 Disj 𝑦𝑌 {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
15 disjss2 4556 . 2 (∀𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} → (Disj 𝑦𝑌 {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} → Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}))
1613, 14, 15mp2 9 1 Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  {cpr 4127  ⟨cop 4131  Disj wdisj 4553  ran crn 5039  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  Word cword 13146   lastS clsw 13147   substr csubstr 13150   WWalks cwwlk 26205   WWalksN cwwlkn 26206 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-rmo 2904  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-disj 4554  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-wwlk 26207  df-wwlkn 26208 This theorem is referenced by:  hashwwlkext  26274
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