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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 29611 | . 2 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) | |
2 | trliswlk 29583 | . 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 5149 ‘cfv 6549 Walkscwlks 29482 Trailsctrls 29576 Pathscpths 29598 |
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 2696 ax-sep 5300 ax-nul 5307 ax-pr 5429 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fv 6557 df-ov 7422 df-wlks 29485 df-trls 29578 df-pths 29602 |
This theorem is referenced by: spthiswlk 29614 pthdadjvtx 29616 2pthnloop 29617 upgr2pthnlp 29618 pthonpth 29634 cycliswlk 29684 wspthsnonn0vne 29800 upgr3v3e3cycl 30062 upgr4cycl4dv4e 30067 pthhashvtx 34868 spthcycl 34870 loop1cycl 34878 |
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