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Mirrors > Home > MPE Home > Th. List > wwlksn | Structured version Visualization version GIF version |
Description: The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.) |
Ref | Expression |
---|---|
wwlksn | ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6901 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺)) | |
2 | 1 | adantl 480 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺)) |
3 | oveq1 7431 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1)) | |
4 | 3 | eqeq2d 2737 | . . . . . 6 ⊢ (𝑛 = 𝑁 → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1))) |
5 | 4 | adantr 479 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1))) |
6 | 2, 5 | rabeqbidv 3437 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}) |
7 | df-wwlksn 29765 | . . . 4 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)}) | |
8 | fvex 6914 | . . . . 5 ⊢ (WWalks‘𝐺) ∈ V | |
9 | 8 | rabex 5339 | . . . 4 ⊢ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V |
10 | 6, 7, 9 | ovmpoa 7581 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}) |
11 | 10 | expcom 412 | . 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})) |
12 | 7 | reldmmpo 7560 | . . . . 5 ⊢ Rel dom WWalksN |
13 | 12 | ovprc2 7464 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅) |
14 | fvprc 6893 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (WWalks‘𝐺) = ∅) | |
15 | 14 | rabeqdv 3435 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)}) |
16 | rab0 4387 | . . . . 5 ⊢ {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅ | |
17 | 15, 16 | eqtrdi 2782 | . . . 4 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅) |
18 | 13, 17 | eqtr4d 2769 | . . 3 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}) |
19 | 18 | a1d 25 | . 2 ⊢ (¬ 𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})) |
20 | 11, 19 | pm2.61i 182 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 ∅c0 4325 ‘cfv 6554 (class class class)co 7424 1c1 11159 + caddc 11161 ℕ0cn0 12524 ♯chash 14347 WWalkscwwlks 29759 WWalksN cwwlksn 29760 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6506 df-fun 6556 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-wwlksn 29765 |
This theorem is referenced by: iswwlksn 29772 wwlksn0s 29795 0enwwlksnge1 29798 wlknwwlksnbij 29822 wwlksnfi 29840 |
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