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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 | β’ (π β (ClWalksβπΊ) β π β (WalksβπΊ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopabran 5568 | . 2 β’ (π β {β¨π, πβ© β£ (π(WalksβπΊ)π β§ (πβ0) = (πβ(β―βπ)))} β π β (WalksβπΊ)) | |
2 | clwlks 29606 | . 2 β’ (ClWalksβπΊ) = {β¨π, πβ© β£ (π(WalksβπΊ)π β§ (πβ0) = (πβ(β―βπ)))} | |
3 | 1, 2 | eleq2s 2847 | 1 β’ (π β (ClWalksβπΊ) β π β (WalksβπΊ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 class class class wbr 5152 {copab 5214 βcfv 6553 0cc0 11146 β―chash 14329 Walkscwlks 29430 ClWalkscclwlks 29604 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-iota 6505 df-fun 6555 df-fv 6561 df-clwlks 29605 |
This theorem is referenced by: clwlkswks 29610 upgrclwlkcompim 29615 clwlkcompbp 29616 clwlkclwwlkflem 29834 clwlknf1oclwwlknlem1 29911 clwlknf1oclwwlkn 29914 clwwlknonclwlknonf1o 30192 dlwwlknondlwlknonf1olem1 30194 wlkl0 30197 |
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