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Mirrors > Home > MPE Home > Th. List > spthson | Structured version Visualization version GIF version |
Description: The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.) |
Ref | Expression |
---|---|
pthsonfval.v | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
spthson | β’ ((π΄ β π β§ π΅ β π) β (π΄(SPathsOnβπΊ)π΅) = {β¨π, πβ© β£ (π(π΄(TrailsOnβπΊ)π΅)π β§ π(SPathsβπΊ)π)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pthsonfval.v | . . . 4 β’ π = (VtxβπΊ) | |
2 | 1 | 1vgrex 28766 | . . 3 β’ (π΄ β π β πΊ β V) |
3 | 2 | adantr 480 | . 2 β’ ((π΄ β π β§ π΅ β π) β πΊ β V) |
4 | simpl 482 | . . 3 β’ ((π΄ β π β§ π΅ β π) β π΄ β π) | |
5 | 4, 1 | eleqtrdi 2837 | . 2 β’ ((π΄ β π β§ π΅ β π) β π΄ β (VtxβπΊ)) |
6 | simpr 484 | . . 3 β’ ((π΄ β π β§ π΅ β π) β π΅ β π) | |
7 | 6, 1 | eleqtrdi 2837 | . 2 β’ ((π΄ β π β§ π΅ β π) β π΅ β (VtxβπΊ)) |
8 | df-spthson 29481 | . 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)})) | |
9 | 3, 5, 7, 8 | mptmpoopabovd 8065 | 1 β’ ((π΄ β π β§ π΅ β π) β (π΄(SPathsOnβπΊ)π΅) = {β¨π, πβ© β£ (π(π΄(TrailsOnβπΊ)π΅)π β§ π(SPathsβπΊ)π)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 class class class wbr 5141 {copab 5203 βcfv 6536 (class class class)co 7404 Vtxcvtx 28760 TrailsOnctrlson 29453 SPathscspths 29475 SPathsOncspthson 29477 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7971 df-2nd 7972 df-spthson 29481 |
This theorem is referenced by: isspthson 29505 |
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