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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 | ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istrl 28473 | . 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | |
2 | 1 | simplbi 498 | 1 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5103 ◡ccnv 5630 Fun wfun 6487 ‘cfv 6493 Walkscwlks 28373 Trailsctrls 28467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fv 6501 df-wlks 28376 df-trls 28469 |
This theorem is referenced by: trlreslem 28476 trlres 28477 trlontrl 28488 pthiswlk 28504 pthdivtx 28506 spthdifv 28510 spthdep 28511 pthdepisspth 28512 usgr2trlspth 28538 crctisclwlk 28571 crctiswlk 28573 crctcshlem3 28593 crctcshwlk 28596 eupthiswlk 28985 eupthres 28988 trlsegvdeglem1 28993 eucrctshift 29016 |
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