|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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 6906 | . . . . . 6 ⊢ (𝑔 = 𝐺 → (WWalks‘𝑔) = (WWalks‘𝐺)) | |
| 2 | 1 | adantl 481 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (WWalks‘𝑔) = (WWalks‘𝐺)) | 
| 3 | oveq1 7438 | . . . . . . 7 ⊢ (𝑛 = 𝑁 → (𝑛 + 1) = (𝑁 + 1)) | |
| 4 | 3 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑛 = 𝑁 → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1))) | 
| 5 | 4 | adantr 480 | . . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → ((♯‘𝑤) = (𝑛 + 1) ↔ (♯‘𝑤) = (𝑁 + 1))) | 
| 6 | 2, 5 | rabeqbidv 3455 | . . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)} = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}) | 
| 7 | df-wwlksn 29851 | . . . 4 ⊢ WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)}) | |
| 8 | fvex 6919 | . . . . 5 ⊢ (WWalks‘𝐺) ∈ V | |
| 9 | 8 | rabex 5339 | . . . 4 ⊢ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ∈ V | 
| 10 | 6, 7, 9 | ovmpoa 7588 | . . 3 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}) | 
| 11 | 10 | expcom 413 | . 2 ⊢ (𝐺 ∈ V → (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})) | 
| 12 | 7 | reldmmpo 7567 | . . . . 5 ⊢ Rel dom WWalksN | 
| 13 | 12 | ovprc2 7471 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (𝑁 WWalksN 𝐺) = ∅) | 
| 14 | fvprc 6898 | . . . . . 6 ⊢ (¬ 𝐺 ∈ V → (WWalks‘𝐺) = ∅) | |
| 15 | 14 | rabeqdv 3452 | . . . . 5 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)}) | 
| 16 | rab0 4386 | . . . . 5 ⊢ {𝑤 ∈ ∅ ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅ | |
| 17 | 15, 16 | eqtrdi 2793 | . . . 4 ⊢ (¬ 𝐺 ∈ V → {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} = ∅) | 
| 18 | 13, 17 | eqtr4d 2780 | . . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∅c0 4333 ‘cfv 6561 (class class class)co 7431 1c1 11156 + caddc 11158 ℕ0cn0 12526 ♯chash 14369 WWalkscwwlks 29845 WWalksN cwwlksn 29846 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-wwlksn 29851 | 
| This theorem is referenced by: iswwlksn 29858 wwlksn0s 29881 0enwwlksnge1 29884 wlknwwlksnbij 29908 wwlksnfi 29926 | 
| Copyright terms: Public domain | W3C validator |