Theorem disjxwwlknOLD 27292

Description: Obsolete version of disjxwwlkn 27291 as of 12-Oct-2022. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)

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
disjxwwlknOLD	⊢ Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ ((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 disjxwwlknOLD

Step	Hyp	Ref	Expression
1		simp1 1127	. . . . . 6 ⊢ (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦)
2	1	rgenw 3105	. . . . 5 ⊢ ∀𝑥 ∈ 𝑋 (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦)
3		ss2rab 3898	. . . . 5 ⊢ ({𝑥 ∈ 𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ 𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ↔ ∀𝑥 ∈ 𝑋 (((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 substr ⟨0, 𝑀⟩) = 𝑦))
4	2, 3	mpbir 223	. . . 4 ⊢ {𝑥 ∈ 𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ 𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
5		wwlksnextprop.x	. . . . . 6 ⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)
6		wwlkssswwlksn 27215	. . . . . . 7 ⊢ ((𝑁 + 1) WWalksN 𝐺) ⊆ (WWalks‘𝐺)
7		eqid 2777	. . . . . . . 8 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
8	7	wwlkssswrd 27211	. . . . . . 7 ⊢ (WWalks‘𝐺) ⊆ Word (Vtx‘𝐺)
9	6, 8	sstri 3829	. . . . . 6 ⊢ ((𝑁 + 1) WWalksN 𝐺) ⊆ Word (Vtx‘𝐺)
10	5, 9	eqsstri 3853	. . . . 5 ⊢ 𝑋 ⊆ Word (Vtx‘𝐺)
11		rabss2 3905	. . . . 5 ⊢ (𝑋 ⊆ Word (Vtx‘𝐺) → {𝑥 ∈ 𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦})
12	10, 11	ax-mp 5	. . . 4 ⊢ {𝑥 ∈ 𝑋 ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
13	4, 12	sstri 3829	. . 3 ⊢ {𝑥 ∈ 𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
14	13	rgenw 3105	. 2 ⊢ ∀𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
15		disjxwrdOLD 13775	. 2 ⊢ Disj 𝑦 ∈ 𝑌 {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦}
16		disjss2 4857	. 2 ⊢ (∀𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} → (Disj 𝑦 ∈ 𝑌 {𝑥 ∈ Word (Vtx‘𝐺) ∣ (𝑥 substr ⟨0, 𝑀⟩) = 𝑦} → Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}))
17	14, 15, 16	mp2 9	1 ⊢ Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}

Syntax hints:   → wi 4   ∧ w3a 1071   = wceq 1601   ∈ wcel 2106  ∀wral 3089  {crab 3093   ⊆ wss 3791  {cpr 4399  ⟨cop 4403  Disj wdisj 4854  ‘cfv 6135  (class class class)co 6922  0cc0 10272  1c1 10273   + caddc 10275  Word cword 13599  lastSclsw 13652   substr csubstr 13730  Vtxcvtx 26344  Edgcedg 26395  WWalkscwwlks 27174   WWalksN cwwlksn 27175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-disj 4855  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-n0 11643  df-z 11729  df-uz 11993  df-fz 12644  df-fzo 12785  df-hash 13436  df-word 13600  df-wwlks 27179  df-wwlksn 27180

This theorem is referenced by:  hashwwlksnextOLD  27294