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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 29462 | . 2 โข (๐น(Trailsโ๐บ)๐ โ (๐น(Walksโ๐บ)๐ โง Fun โก๐น)) | |
2 | 1 | simplbi 497 | 1 โข (๐น(Trailsโ๐บ)๐ โ ๐น(Walksโ๐บ)๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 class class class wbr 5141 โกccnv 5668 Fun wfun 6531 โcfv 6537 Walkscwlks 29362 Trailsctrls 29456 |
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-pr 5420 |
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-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 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 6489 df-fun 6539 df-fv 6545 df-wlks 29365 df-trls 29458 |
This theorem is referenced by: trlreslem 29465 trlres 29466 trlontrl 29477 pthiswlk 29493 pthdivtx 29495 spthdifv 29499 spthdep 29500 pthdepisspth 29501 usgr2trlspth 29527 crctisclwlk 29560 crctiswlk 29562 crctcshlem3 29582 crctcshwlk 29585 eupthiswlk 29974 eupthres 29977 trlsegvdeglem1 29982 eucrctshift 30005 |
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