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Mirrors > Home > MPE Home > Th. List > clwlkwlk | Structured version Visualization version GIF version |
Description: Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
clwlkwlk | ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopabran 5558 | . 2 ⊢ (𝑊 ∈ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))} → 𝑊 ∈ (Walks‘𝐺)) | |
2 | clwlks 29703 | . 2 ⊢ (ClWalks‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))} | |
3 | 1, 2 | eleq2s 2844 | 1 ⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 class class class wbr 5143 {copab 5205 ‘cfv 6543 0cc0 11146 ♯chash 14339 Walkscwlks 29527 ClWalkscclwlks 29701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6495 df-fun 6545 df-fv 6551 df-clwlks 29702 |
This theorem is referenced by: clwlkswks 29707 upgrclwlkcompim 29712 clwlkcompbp 29713 clwlkclwwlkflem 29931 clwlknf1oclwwlknlem1 30008 clwlknf1oclwwlkn 30011 clwwlknonclwlknonf1o 30289 dlwwlknondlwlknonf1olem1 30291 wlkl0 30294 |
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