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Mirrors > Home > MPE Home > Th. List > iswwlksn | Structured version Visualization version GIF version |
Description: A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.) |
Ref | Expression |
---|---|
iswwlksn | ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wwlksn 27777 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)}) | |
2 | 1 | eleq2d 2818 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})) |
3 | fveqeq2 6685 | . . 3 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = (𝑁 + 1) ↔ (♯‘𝑊) = (𝑁 + 1))) | |
4 | 3 | elrab 3588 | . 2 ⊢ (𝑊 ∈ {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)} ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1))) |
5 | 2, 4 | bitrdi 290 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 {crab 3057 ‘cfv 6339 (class class class)co 7172 1c1 10618 + caddc 10620 ℕ0cn0 11978 ♯chash 13784 WWalkscwwlks 27765 WWalksN cwwlksn 27766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pr 5296 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-iota 6297 df-fun 6341 df-fv 6347 df-ov 7175 df-oprab 7176 df-mpo 7177 df-wwlksn 27771 |
This theorem is referenced by: wwlksnprcl 27779 iswwlksnx 27780 wwlknbp 27782 wwlknp 27783 wwlkswwlksn 27805 wlklnwwlkln1 27808 wlklnwwlkln2lem 27822 wlknewwlksn 27827 wwlksnred 27832 wwlksnext 27833 wwlksnextproplem3 27851 wspthsnonn0vne 27857 elwspths2spth 27907 rusgrnumwwlkl1 27908 clwwlkel 27985 clwwlkf 27986 clwwlknwwlksnb 27994 |
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