Theorem disjxwwlks 26264

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.)

Assertion

Ref	Expression
disjxwwlks	⊢ Disj 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}

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

Allowed substitution hints:   𝑃(𝑥,𝑦)

Proof of Theorem disjxwwlks

Step	Hyp	Ref	Expression
1		simp1 1054	. . . . 5 ⊢ (((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) → (𝑥 substr ⟨0, 𝑁⟩) = 𝑦)
2	1	a1i 11	. . . 4 ⊢ (𝑥 ∈ Word 𝑉 → (((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸) → (𝑥 substr ⟨0, 𝑁⟩) = 𝑦))
3	2	ss2rabi 3647	. . 3 ⊢ {𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}
4	3	rgenw 2908	. 2 ⊢ ∀𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}
5		disjxwrd 13307	. 2 ⊢ Disj 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦}
6		disjss2 4556	. 2 ⊢ (∀𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦} → (Disj 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ (𝑥 substr ⟨0, 𝑁⟩) = 𝑦} → Disj 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}))
7	4, 5, 6	mp2 9	1 ⊢ Disj 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}

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  Word cword 13146   lastS clsw 13147   substr csubstr 13150   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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  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-ral 2901  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-in 3547  df-ss 3554  df-disj 4554

This theorem is referenced by: (None)