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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 29870 | . . 3 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) | |
| 2 | 1 | simp2d 1144 | . 2 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → 𝑤 ∈ (𝑁 WWalksN 𝐺)) |
| 3 | 2 | ssriv 3987 | 1 ⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℕ0cn0 12526 SPathscspths 29731 WWalksN cwwlksn 29846 WSPathsN cwwspthsn 29848 |
| 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 df-wspthsn 29853 |
| This theorem is referenced by: wspthnfi 29939 fusgreg2wsp 30355 |
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