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Mirrors > Home > MPE Home > Th. List > 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.) (Revised by AV, 27-Oct-2022.) |
Ref | Expression |
---|---|
wwlksnexthasheq.v | ⊢ 𝑉 = (Vtx‘𝐺) |
wwlksnexthasheq.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
disjxwwlksn | ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . . . 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 4087 | . . 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 3063 | . 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 14735 | . 2 ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦} | |
6 | disjss2 5118 | . 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)} |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 {cpr 4633 Disj wdisj 5115 ‘cfv 6563 (class class class)co 7431 0cc0 11153 Word cword 14549 lastSclsw 14597 prefix cpfx 14705 Vtxcvtx 29028 Edgcedg 29079 WWalksN cwwlksn 29856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rmo 3378 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-ss 3980 df-disj 5116 |
This theorem is referenced by: (None) |
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