![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > spthonprop | Structured version Visualization version GIF version |
Description: Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) |
Ref | Expression |
---|---|
pthsonfval.v | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
spthonprop | β’ (πΉ(π΄(SPathsOnβπΊ)π΅)π β ((πΊ β V β§ π΄ β π β§ π΅ β π) β§ (πΉ β V β§ π β V) β§ (πΉ(π΄(TrailsOnβπΊ)π΅)π β§ πΉ(SPathsβπΊ)π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pthsonfval.v | . 2 β’ π = (VtxβπΊ) | |
2 | 1 | isspthson 29000 | . . 3 β’ (((π΄ β π β§ π΅ β π) β§ (πΉ β V β§ π β V)) β (πΉ(π΄(SPathsOnβπΊ)π΅)π β (πΉ(π΄(TrailsOnβπΊ)π΅)π β§ πΉ(SPathsβπΊ)π))) |
3 | 2 | 3adantl1 1167 | . 2 β’ (((πΊ β V β§ π΄ β π β§ π΅ β π) β§ (πΉ β V β§ π β V)) β (πΉ(π΄(SPathsOnβπΊ)π΅)π β (πΉ(π΄(TrailsOnβπΊ)π΅)π β§ πΉ(SPathsβπΊ)π))) |
4 | df-spthson 28976 | . 2 β’ SPathsOn = (π β V β¦ (π β (Vtxβπ), π β (Vtxβπ) β¦ {β¨π, πβ© β£ (π(π(TrailsOnβπ)π)π β§ π(SPathsβπ)π)})) | |
5 | 1, 3, 4 | wksonproplem 28961 | 1 β’ (πΉ(π΄(SPathsOnβπΊ)π΅)π β ((πΊ β V β§ π΄ β π β§ π΅ β π) β§ (πΉ β V β§ π β V) β§ (πΉ(π΄(TrailsOnβπΊ)π΅)π β§ πΉ(SPathsβπΊ)π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3475 class class class wbr 5149 βcfv 6544 (class class class)co 7409 Vtxcvtx 28256 TrailsOnctrlson 28948 SPathscspths 28970 SPathsOncspthson 28972 |
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-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
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-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 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-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 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-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-spthson 28976 |
This theorem is referenced by: spthonisspth 29007 spthonpthon 29008 spthonepeq 29009 |
Copyright terms: Public domain | W3C validator |