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Mirrors > Home > MPE Home > Th. List > isspth | Structured version Visualization version GIF version |
Description: Conditions for a pair of classes/functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.) |
Ref | Expression |
---|---|
isspth | β’ (πΉ(SPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ Fun β‘π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | spthsfval 28378 | . 2 β’ (SPathsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ Fun β‘π)} | |
2 | cnveq 5815 | . . . 4 β’ (π = π β β‘π = β‘π) | |
3 | 2 | funeqd 6506 | . . 3 β’ (π = π β (Fun β‘π β Fun β‘π)) |
4 | 3 | adantl 482 | . 2 β’ ((π = πΉ β§ π = π) β (Fun β‘π β Fun β‘π)) |
5 | reltrls 28350 | . 2 β’ Rel (TrailsβπΊ) | |
6 | 1, 4, 5 | brfvopabrbr 6928 | 1 β’ (πΉ(SPathsβπΊ)π β (πΉ(TrailsβπΊ)π β§ Fun β‘π)) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 β§ wa 396 = wceq 1540 class class class wbr 5092 β‘ccnv 5619 Fun wfun 6473 βcfv 6479 Trailsctrls 28346 SPathscspths 28369 |
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-sep 5243 ax-nul 5250 ax-pr 5372 |
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-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-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 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-fv 6487 df-trls 28348 df-spths 28373 |
This theorem is referenced by: spthispth 28382 spthdifv 28389 spthdep 28390 pthdepisspth 28391 spthonepeq 28408 uhgrwkspth 28411 usgr2wlkspth 28415 usgr2pth 28420 2spthd 28594 0spth 28778 3spthd 28828 spthcycl 33390 |
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