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Mirrors > Home > MPE Home > Th. List > trliswlk | Structured version Visualization version GIF version |
Description: A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 29-Oct-2021.) |
Ref | Expression |
---|---|
trliswlk | โข (๐น(Trailsโ๐บ)๐ โ ๐น(Walksโ๐บ)๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istrl 29538 | . 2 โข (๐น(Trailsโ๐บ)๐ โ (๐น(Walksโ๐บ)๐ โง Fun โก๐น)) | |
2 | 1 | simplbi 496 | 1 โข (๐น(Trailsโ๐บ)๐ โ ๐น(Walksโ๐บ)๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 class class class wbr 5152 โกccnv 5681 Fun wfun 6547 โcfv 6553 Walkscwlks 29438 Trailsctrls 29532 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fv 6561 df-wlks 29441 df-trls 29534 |
This theorem is referenced by: trlreslem 29541 trlres 29542 trlontrl 29553 pthiswlk 29569 pthdivtx 29571 spthdifv 29575 spthdep 29576 pthdepisspth 29577 usgr2trlspth 29603 crctisclwlk 29636 crctiswlk 29638 crctcshlem3 29658 crctcshwlk 29661 eupthiswlk 30050 eupthres 30053 trlsegvdeglem1 30058 eucrctshift 30081 |
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