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Mirrors > Home > MPE Home > Th. List > Mathboxes > disjxwwlksn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
wwlksnexthasheq.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wwlksnexthasheq.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
disjxwwlksn | ⊢ Disj 𝑦 ∈ (𝑁 WWalkSN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr 〈0, 𝑁〉) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . . . . 5 ⊢ (((𝑥 substr 〈0, 𝑁〉) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 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)} ∈ 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)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr 〈0, 𝑁〉) = 𝑦} |
4 | 3 | rgenw 2908 | . 2 ⊢ ∀𝑦 ∈ (𝑁 WWalkSN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr 〈0, 𝑁〉) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr 〈0, 𝑁〉) = 𝑦} |
5 | disjxwrd 13307 | . 2 ⊢ Disj 𝑦 ∈ (𝑁 WWalkSN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 substr 〈0, 𝑁〉) = 𝑦} | |
6 | disjss2 4556 | . 2 ⊢ (∀𝑦 ∈ (𝑁 WWalkSN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr 〈0, 𝑁〉) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} ⊆ {𝑥 ∈ Word 𝑉 ∣ (𝑥 substr 〈0, 𝑁〉) = 𝑦} → (Disj 𝑦 ∈ (𝑁 WWalkSN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 substr 〈0, 𝑁〉) = 𝑦} → Disj 𝑦 ∈ (𝑁 WWalkSN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr 〈0, 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 substr 〈0, 𝑁〉) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 {crab 2900 ⊆ wss 3540 {cpr 4127 〈cop 4131 Disj wdisj 4553 ‘cfv 5804 (class class class)co 6549 0cc0 9815 Word cword 13146 lastS clsw 13147 substr csubstr 13150 Vtxcvtx 25673 Edgcedga 25792 WWalkSN cwwlksn 41029 |
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) |
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