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Mirrors > Home > MPE Home > Th. List > relwlk | Structured version Visualization version GIF version |
Description: The set (WalksβπΊ) of all walks on πΊ is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.) |
Ref | Expression |
---|---|
relwlk | β’ Rel (WalksβπΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-wlks 29111 | . 2 β’ Walks = (π β V β¦ {β¨π, πβ© β£ (π β Word dom (iEdgβπ) β§ π:(0...(β―βπ))βΆ(Vtxβπ) β§ βπ β (0..^(β―βπ))if-((πβπ) = (πβ(π + 1)), ((iEdgβπ)β(πβπ)) = {(πβπ)}, {(πβπ), (πβ(π + 1))} β ((iEdgβπ)β(πβπ))))}) | |
2 | 1 | relmptopab 7658 | 1 β’ Rel (WalksβπΊ) |
Colors of variables: wff setvar class |
Syntax hints: if-wif 1061 β§ w3a 1087 = wceq 1541 β wcel 2106 βwral 3061 Vcvv 3474 β wss 3948 {csn 4628 {cpr 4630 dom cdm 5676 Rel wrel 5681 βΆwf 6539 βcfv 6543 (class class class)co 7411 0cc0 11112 1c1 11113 + caddc 11115 ...cfz 13488 ..^cfzo 13631 β―chash 14294 Word cword 14468 Vtxcvtx 28511 iEdgciedg 28512 Walkscwlks 29108 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-wlks 29111 |
This theorem is referenced by: wlkop 29140 istrl 29208 isclwlk 29285 usgrgt2cycl 34407 |
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