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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 26737 | . . 3 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑤 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑤)) | |
2 | 1 | simp2d 1074 | . 2 ⊢ (𝑤 ∈ (𝑁 WSPathsN 𝐺) → 𝑤 ∈ (𝑁 WWalksN 𝐺)) |
3 | 2 | ssriv 3607 | 1 ⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 ⊆ wss 3574 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 ℕ0cn0 11292 SPathscspths 26609 WWalksN cwwlksn 26718 WSPathsN cwwspthsn 26720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-wwlksn 26723 df-wspthsn 26725 |
This theorem is referenced by: wspthnfi 26815 fusgreg2wsp 27200 |
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