Theorem disjxwwlksn 29886

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, 29-Jul-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 27-Oct-2022.)

Hypotheses

Ref	Expression
wwlksnexthasheq.v	⊢ 𝑉 = (Vtx‘𝐺)
wwlksnexthasheq.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
disjxwwlksn	⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}

Distinct variable groups:   𝑦,𝑁   𝑥,𝑉   𝑥,𝑦

Allowed substitution hints:   𝑃(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑁(𝑥)   𝑉(𝑦)

Proof of Theorem disjxwwlksn

Step	Hyp	Ref	Expression
1		simp1 1136	. . . . 5 ⊢ (((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 prefix 𝑁) = 𝑦)
2	1	a1i 11	. . . 4 ⊢ (𝑥 ∈ Word 𝑉 → (((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸) → (𝑥 prefix 𝑁) = 𝑦))
3	2	ss2rabi 4036	. . 3 ⊢ {𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
4	3	rgenw 3048	. 2 ⊢ ∀𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
5		disjwrdpfx 14643	. 2 ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
6		disjss2 5072	. 2 ⊢ (∀𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦} → (Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦} → Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}))
7	4, 5, 6	mp2 9	1 ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402   ⊆ wss 3911  {cpr 4587  Disj wdisj 5069  ‘cfv 6500  (class class class)co 7370  0cc0 11047  Word cword 14457  lastSclsw 14506   prefix cpfx 14614  Vtxcvtx 28978  Edgcedg 29029   WWalksN cwwlksn 29808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rmo 3351  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-ss 3928  df-disj 5070

This theorem is referenced by: (None)