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| Mirrors > Home > MPE Home > Th. List > reltrls | Structured version Visualization version GIF version | ||
| Description: The set (Trails‘𝐺) of all trails on 𝐺 is a set of pairs by our definition of a trail, and so is a relation. (Contributed by AV, 29-Oct-2021.) |
| Ref | Expression |
|---|---|
| reltrls | ⊢ Rel (Trails‘𝐺) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-trls 29710 | . 2 ⊢ Trails = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}) | |
| 2 | 1 | relmptopab 7683 | 1 ⊢ Rel (Trails‘𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 Vcvv 3480 class class class wbr 5143 ◡ccnv 5684 Rel wrel 5690 Fun wfun 6555 ‘cfv 6561 Walkscwlks 29614 Trailsctrls 29708 |
| 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-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-trls 29710 |
| This theorem is referenced by: ispth 29741 isspth 29742 iscrct 29810 iseupth 30220 |
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