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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 29549 | . 2 β’ (SPathsβπΊ) = {β¨π, πβ© β£ (π(TrailsβπΊ)π β§ Fun β‘π)} | |
2 | cnveq 5876 | . . . 4 β’ (π = π β β‘π = β‘π) | |
3 | 2 | funeqd 6575 | . . 3 β’ (π = π β (Fun β‘π β Fun β‘π)) |
4 | 3 | adantl 481 | . 2 β’ ((π = πΉ β§ π = π) β (Fun β‘π β Fun β‘π)) |
5 | reltrls 29521 | . 2 β’ Rel (TrailsβπΊ) | |
6 | 1, 4, 5 | brfvopabrbr 7002 | 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 395 = wceq 1534 class class class wbr 5148 β‘ccnv 5677 Fun wfun 6542 βcfv 6548 Trailsctrls 29517 SPathscspths 29540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-trls 29519 df-spths 29544 |
This theorem is referenced by: spthispth 29553 spthdifv 29560 spthdep 29561 pthdepisspth 29562 spthonepeq 29579 uhgrwkspth 29582 usgr2wlkspth 29586 usgr2pth 29591 2spthd 29765 0spth 29949 3spthd 29999 spthcycl 34739 |
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