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Mirrors > Home > MPE Home > Th. List > wwlksonvtx | Structured version Visualization version GIF version |
Description: If a word 𝑊 represents a walk of length 2 on two classes 𝐴 and 𝐶, these classes are vertices. (Contributed by AV, 14-Mar-2022.) |
Ref | Expression |
---|---|
wwlksonvtx.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
wwlksonvtx | ⊢ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6920 | . . . . 5 ⊢ (Vtx‘𝑔) ∈ V | |
2 | 1, 1 | pm3.2i 470 | . . . 4 ⊢ ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V) |
3 | 2 | rgen2w 3064 | . . 3 ⊢ ∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V) |
4 | df-wwlksnon 29862 | . . . 4 ⊢ WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)})) | |
5 | fveq2 6907 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
6 | 5, 5 | jca 511 | . . . . 5 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺))) |
7 | 6 | adantl 481 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((Vtx‘𝑔) = (Vtx‘𝐺) ∧ (Vtx‘𝑔) = (Vtx‘𝐺))) |
8 | 4, 7 | el2mpocl 8110 | . . 3 ⊢ (∀𝑛 ∈ ℕ0 ∀𝑔 ∈ V ((Vtx‘𝑔) ∈ V ∧ (Vtx‘𝑔) ∈ V) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))) |
9 | 3, 8 | ax-mp 5 | . 2 ⊢ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)))) |
10 | wwlksonvtx.v | . . . . 5 ⊢ 𝑉 = (Vtx‘𝐺) | |
11 | 10 | eleq2i 2831 | . . . 4 ⊢ (𝐴 ∈ 𝑉 ↔ 𝐴 ∈ (Vtx‘𝐺)) |
12 | 10 | eleq2i 2831 | . . . 4 ⊢ (𝐶 ∈ 𝑉 ↔ 𝐶 ∈ (Vtx‘𝐺)) |
13 | 11, 12 | anbi12i 628 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) ↔ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))) |
14 | 13 | biimpri 228 | . 2 ⊢ ((𝐴 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺)) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
15 | 9, 14 | simpl2im 503 | 1 ⊢ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 Vcvv 3478 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ℕ0cn0 12524 Vtxcvtx 29028 WWalksN cwwlksn 29856 WWalksNOn cwwlksnon 29857 |
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 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-wwlksnon 29862 |
This theorem is referenced by: iswspthsnon 29886 wwlks2onv 29983 elwwlks2ons3im 29984 |
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