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| Mirrors > Home > MPE Home > Th. List > pthistrl | Structured version Visualization version GIF version | ||
| Description: A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.) |
| Ref | Expression |
|---|---|
| pthistrl | ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ispth 29741 | . 2 ⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹))) ∧ ((𝑃 “ {0, (♯‘𝐹)}) ∩ (𝑃 “ (1..^(♯‘𝐹)))) = ∅)) | |
| 2 | 1 | simp1bi 1146 | 1 ⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∩ cin 3950 ∅c0 4333 {cpr 4628 class class class wbr 5143 ◡ccnv 5684 ↾ cres 5687 “ cima 5688 Fun wfun 6555 ‘cfv 6561 (class class class)co 7431 0cc0 11155 1c1 11156 ..^cfzo 13694 ♯chash 14369 Trailsctrls 29708 Pathscpths 29730 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-trls 29710 df-pths 29734 |
| This theorem is referenced by: pthiswlk 29745 pthonpth 29768 isspthonpth 29769 usgr2trlspth 29781 usgr2pthspth 29782 cycliscrct 29819 spthcycl 35134 |
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