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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 29728 | . 2 ⊢ Trails = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}) | |
2 | 1 | relmptopab 7700 | 1 ⊢ Rel (Trails‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 Vcvv 3488 class class class wbr 5166 ◡ccnv 5699 Rel wrel 5705 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-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-trls 29728 |
This theorem is referenced by: ispth 29759 isspth 29760 iscrct 29826 iseupth 30233 |
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