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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 29732 | . 2 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹)) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5166 ◡ccnv 5699 Fun wfun 6567 ‘cfv 6573 Walkscwlks 29632 Trailsctrls 29726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-br 5167 df-opab 5229 df-mpt 5250 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-iota 6525 df-fun 6575 df-fv 6581 df-wlks 29635 df-trls 29728 |
This theorem is referenced by: trlreslem 29735 trlres 29736 trlontrl 29747 pthiswlk 29763 pthdivtx 29765 spthdifv 29769 spthdep 29770 pthdepisspth 29771 usgr2trlspth 29797 crctisclwlk 29830 crctiswlk 29832 crctcshlem3 29852 crctcshwlk 29855 eupthiswlk 30244 eupthres 30247 trlsegvdeglem1 30252 eucrctshift 30275 |
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