Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > 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 41040 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (𝑁 WWalkSN 𝐺) = {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)}) | |
2 | 1 | eleq2d 2673 | . 2 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalkSN 𝐺) ↔ 𝑊 ∈ {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})) |
3 | fveq2 6103 | . . . 4 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊)) | |
4 | 3 | eqeq1d 2612 | . . 3 ⊢ (𝑤 = 𝑊 → ((#‘𝑤) = (𝑁 + 1) ↔ (#‘𝑊) = (𝑁 + 1))) |
5 | 4 | elrab 3331 | . 2 ⊢ (𝑊 ∈ {𝑤 ∈ (WWalkS‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)} ↔ (𝑊 ∈ (WWalkS‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))) |
6 | 2, 5 | syl6bb 275 | 1 ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalkSN 𝐺) ↔ (𝑊 ∈ (WWalkS‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {crab 2900 ‘cfv 5804 (class class class)co 6549 1c1 9816 + caddc 9818 ℕ0cn0 11169 #chash 12979 WWalkScwwlks 41028 WWalkSN cwwlksn 41029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-wwlksn 41034 |
This theorem is referenced by: iswwlksnx 41042 wwlknbp 41044 wwlknp 41045 wwlkswwlksn 41061 1wlklnwwlkln1 41065 1wlklnwwlkln2lem 41079 wlknewwlksn 41084 wwlksnred 41098 wwlksnext 41099 wwlksnextproplem3 41117 wspthsnonn0vne 41124 elwspths2spth 41171 rusgrnumwwlkl1 41172 clwwlksel 41221 clwwlksf 41222 |
Copyright terms: Public domain | W3C validator |