Theorem disjxwwlksn 30523

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 1154	. . . . 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 4004	. . 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 3084	. 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 14877	. 2 ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦}
6		disjss2 5076	. 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)}

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  {crab 3416   ⊆ wss 3900  {cpr 4590  Disj wdisj 5073  ‘cfv 6558  (class class class)co 7443  0cc0 11227  Word cword 14685  lastSclsw 14734   prefix cpfx 14848  Vtxcvtx 29604  Edgcedg 29655   WWalksN cwwlksn 30445

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rmo 3369  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-disj 5074

This theorem is used by: (None)