Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > wspthsswwlknon | 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 between the two vertices. (Contributed by AV, 15-May-2021.) |
Ref | Expression |
---|---|
wspthsswwlknon | ⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ⊆ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2610 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
2 | 1 | wspthnonp 41055 | . . 3 ⊢ (𝑤 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤))) |
3 | simp3l 1082 | . . 3 ⊢ (((𝑁 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺)) ∧ (𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤)) → 𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝑤 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → 𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)) |
5 | 4 | ssriv 3572 | 1 ⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ⊆ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∧ w3a 1031 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 ⊆ wss 3540 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℕ0cn0 11169 Vtxcvtx 25673 SPathsOncspthson 40922 WWalksNOn cwwlksnon 41030 WSPathsNOn cwwspthsnon 41032 |
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-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 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-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-wwlksnon 41035 df-wspthsnon 41037 |
This theorem is referenced by: wspthnonfi 41129 |
Copyright terms: Public domain | W3C validator |