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Theorem disjxwwlkn 27705
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.) (Revised by AV, 26-Oct-2022.)
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 prefix 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 1133 . . . . . 6 (((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 prefix 𝑀) = 𝑦)
21rgenw 3145 . . . . 5 𝑥𝑋 (((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 prefix 𝑀) = 𝑦)
3 ss2rab 4033 . . . . 5 ({𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥𝑋 ∣ (𝑥 prefix 𝑀) = 𝑦} ↔ ∀𝑥𝑋 (((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 prefix 𝑀) = 𝑦))
42, 3mpbir 234 . . . 4 {𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥𝑋 ∣ (𝑥 prefix 𝑀) = 𝑦}
5 wwlksnextprop.x . . . . . 6 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
6 wwlkssswwlksn 27658 . . . . . . 7 ((𝑁 + 1) WWalksN 𝐺) ⊆ (WWalks‘𝐺)
7 eqid 2824 . . . . . . . 8 (Vtx‘𝐺) = (Vtx‘𝐺)
87wwlkssswrd 27654 . . . . . . 7 (WWalks‘𝐺) ⊆ Word (Vtx‘𝐺)
96, 8sstri 3962 . . . . . 6 ((𝑁 + 1) WWalksN 𝐺) ⊆ Word (Vtx‘𝐺)
105, 9eqsstri 3987 . . . . 5 𝑋 ⊆ Word (Vtx‘𝐺)
11 rabss2 4040 . . . . 5 (𝑋 ⊆ Word (Vtx‘𝐺) → {𝑥𝑋 ∣ (𝑥 prefix 𝑀) = 𝑦} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 prefix 𝑀) = 𝑦})
1210, 11ax-mp 5 . . . 4 {𝑥𝑋 ∣ (𝑥 prefix 𝑀) = 𝑦} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 prefix 𝑀) = 𝑦}
134, 12sstri 3962 . . 3 {𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 prefix 𝑀) = 𝑦}
1413rgenw 3145 . 2 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 prefix 𝑀) = 𝑦}
15 disjwrdpfx 14062 . 2 Disj 𝑦𝑌 {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 prefix 𝑀) = 𝑦}
16 disjss2 5020 . 2 (∀𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 prefix 𝑀) = 𝑦} → (Disj 𝑦𝑌 {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 prefix 𝑀) = 𝑦} → Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}))
1714, 15, 16mp2 9 1 Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  wral 3133  {crab 3137  wss 3919  {cpr 4552  Disj wdisj 5017  cfv 6343  (class class class)co 7149  0cc0 10535  1c1 10536   + caddc 10538  Word cword 13866  lastSclsw 13914   prefix cpfx 14032  Vtxcvtx 26795  Edgcedg 26846  WWalkscwwlks 27617   WWalksN cwwlksn 27618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-disj 5018  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-1st 7684  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-1o 8098  df-er 8285  df-map 8404  df-en 8506  df-dom 8507  df-sdom 8508  df-fin 8509  df-card 9365  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-n0 11895  df-z 11979  df-uz 12241  df-fz 12895  df-fzo 13038  df-hash 13696  df-word 13867  df-wwlks 27622  df-wwlksn 27623
This theorem is referenced by:  hashwwlksnext  27706
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