Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pthsonfval | Structured version Visualization version GIF version |
Description: The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.) |
Ref | Expression |
---|---|
pthsonfval.v | β’ π = (VtxβπΊ) |
Ref | Expression |
---|---|
pthsonfval | β’ ((π΄ β π β§ π΅ β π) β (π΄(PathsOnβπΊ)π΅) = {β¨π, πβ© β£ (π(π΄(TrailsOnβπΊ)π΅)π β§ π(PathsβπΊ)π)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pthsonfval.v | . . . 4 β’ π = (VtxβπΊ) | |
2 | 1 | 1vgrex 27661 | . . 3 β’ (π΄ β π β πΊ β V) |
3 | 2 | adantr 481 | . 2 β’ ((π΄ β π β§ π΅ β π) β πΊ β V) |
4 | simpl 483 | . . 3 β’ ((π΄ β π β§ π΅ β π) β π΄ β π) | |
5 | 4, 1 | eleqtrdi 2847 | . 2 β’ ((π΄ β π β§ π΅ β π) β π΄ β (VtxβπΊ)) |
6 | simpr 485 | . . 3 β’ ((π΄ β π β§ π΅ β π) β π΅ β π) | |
7 | 6, 1 | eleqtrdi 2847 | . 2 β’ ((π΄ β π β§ π΅ β π) β π΅ β (VtxβπΊ)) |
8 | df-pthson 28374 | . 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)})) | |
9 | 3, 5, 7, 8 | mptmpoopabovd 7990 | 1 β’ ((π΄ β π β§ π΅ β π) β (π΄(PathsOnβπΊ)π΅) = {β¨π, πβ© β£ (π(π΄(TrailsOnβπΊ)π΅)π β§ π(PathsβπΊ)π)}) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 396 = wceq 1540 β wcel 2105 Vcvv 3441 class class class wbr 5092 {copab 5154 βcfv 6479 (class class class)co 7337 Vtxcvtx 27655 TrailsOnctrlson 28347 Pathscpths 28368 PathsOncpthson 28370 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-id 5518 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-oprab 7341 df-mpo 7342 df-1st 7899 df-2nd 7900 df-pthson 28374 |
This theorem is referenced by: ispthson 28398 |
Copyright terms: Public domain | W3C validator |