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Mirrors > Home > MPE Home > Th. List > pthiswlk | Structured version Visualization version GIF version |
Description: A path is a walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.) |
Ref | Expression |
---|---|
pthiswlk | ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pthistrl 29758 | . 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
2 | trliswlk 29730 | . 2 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 class class class wbr 5148 ‘cfv 6563 Walkscwlks 29629 Trailsctrls 29723 Pathscpths 29745 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fv 6571 df-ov 7434 df-wlks 29632 df-trls 29725 df-pths 29749 |
This theorem is referenced by: spthiswlk 29761 pthdadjvtx 29763 2pthnloop 29764 upgr2pthnlp 29765 pthonpth 29781 cycliswlk 29831 wspthsnonn0vne 29947 upgr3v3e3cycl 30209 upgr4cycl4dv4e 30214 pthhashvtx 35112 spthcycl 35114 loop1cycl 35122 |
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