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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 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 4034 | . . 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 3068 | . 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 14587 | . 2 ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ (𝑥 prefix 𝑁) = 𝑦} | |
6 | disjss2 5073 | . 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 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 {crab 3407 ⊆ wss 3910 {cpr 4588 Disj wdisj 5070 ‘cfv 6496 (class class class)co 7356 0cc0 11050 Word cword 14401 lastSclsw 14449 prefix cpfx 14557 Vtxcvtx 27894 Edgcedg 27945 WWalksN cwwlksn 28718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ral 3065 df-rmo 3353 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-in 3917 df-ss 3927 df-disj 5071 |
This theorem is referenced by: (None) |
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