![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pthsonprop | Structured version Visualization version GIF version |
Description: Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 16-Jan-2021.) |
Ref | Expression |
---|---|
pthsonfval.v | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
pthsonprop | β’ (πΉ(π΄(PathsOnβπΊ)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(PathsβπΊ)π))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pthsonfval.v | . 2 β’ π = (VtxβπΊ) | |
2 | 1 | ispthson 28996 | . . 3 β’ (((π΄ β π β§ π΅ β π) β§ (πΉ β V β§ π β V)) β (πΉ(π΄(PathsOnβπΊ)π΅)π β (πΉ(π΄(TrailsOnβπΊ)π΅)π β§ πΉ(PathsβπΊ)π))) |
3 | 2 | 3adantl1 1166 | . 2 β’ (((πΊ β V β§ π΄ β π β§ π΅ β π) β§ (πΉ β V β§ π β V)) β (πΉ(π΄(PathsOnβπΊ)π΅)π β (πΉ(π΄(TrailsOnβπΊ)π΅)π β§ πΉ(PathsβπΊ)π))) |
4 | df-pthson 28972 | . 2 β’ PathsOn = (π β V β¦ (π β (Vtxβπ), π β (Vtxβπ) β¦ {β¨π, πβ© β£ (π(π(TrailsOnβπ)π)π β§ π(Pathsβπ)π)})) | |
5 | 1, 3, 4 | wksonproplem 28958 | 1 β’ (πΉ(π΄(PathsOnβπΊ)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(PathsβπΊ)π))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 396 β§ w3a 1087 = wceq 1541 β wcel 2106 Vcvv 3474 class class class wbr 5148 βcfv 6543 (class class class)co 7408 Vtxcvtx 28253 TrailsOnctrlson 28945 Pathscpths 28966 PathsOncpthson 28968 |
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-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 |
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-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 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-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 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-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-pthson 28972 |
This theorem is referenced by: pthonispth 29000 pthontrlon 29001 |
Copyright terms: Public domain | W3C validator |