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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 28856 | . 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 7656 | 1 β’ Rel (WalksβπΊ) |
Colors of variables: wff setvar class |
Syntax hints: if-wif 1062 β§ w3a 1088 = wceq 1542 β wcel 2107 βwral 3062 Vcvv 3475 β wss 3949 {csn 4629 {cpr 4631 dom cdm 5677 Rel wrel 5682 βΆwf 6540 βcfv 6544 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 ...cfz 13484 ..^cfzo 13627 β―chash 14290 Word cword 14464 Vtxcvtx 28256 iEdgciedg 28257 Walkscwlks 28853 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fv 6552 df-wlks 28856 |
This theorem is referenced by: wlkop 28885 istrl 28953 isclwlk 29030 usgrgt2cycl 34121 |
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