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Mirrors > Home > MPE Home > Th. List > wspthsswwlkn | Structured version Visualization version GIF version |
Description: The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length. (Contributed by AV, 18-May-2021.) |
Ref | Expression |
---|---|
wspthsswwlkn | ⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wspthnp 29880 | . . 3 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) | |
2 | 1 | simp2d 1142 | . 2 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → 𝑤 ∈ (𝑁 WWalksN 𝐺)) |
3 | 2 | ssriv 3999 | 1 ⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℕ0cn0 12524 SPathscspths 29746 WWalksN cwwlksn 29856 WSPathsN cwwspthsn 29858 |
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-sep 5302 ax-nul 5312 ax-pr 5438 |
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-rab 3434 df-v 3480 df-sbc 3792 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-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-wwlksn 29861 df-wspthsn 29863 |
This theorem is referenced by: wspthnfi 29949 fusgreg2wsp 30365 |
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