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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 27199 | . . 3 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) | |
2 | 1 | simp2d 1134 | . 2 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → 𝑤 ∈ (𝑁 WWalksN 𝐺)) |
3 | 2 | ssriv 3825 | 1 ⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∃wex 1823 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 class class class wbr 4886 ‘cfv 6135 (class class class)co 6922 ℕ0cn0 11642 SPathscspths 27065 WWalksN cwwlksn 27175 WSPathsN cwwspthsn 27177 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-wwlksn 27180 df-wspthsn 27182 |
This theorem is referenced by: wspthnfi 27299 fusgreg2wsp 27744 |
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