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Mirrors > Home > MPE Home > Th. List > Mathboxes > releupth | Structured version Visualization version GIF version |
Description: The set (EulerPaths‘𝐺) of all Eulerian paths on 𝐺 is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.) |
Ref | Expression |
---|---|
releupth | ⊢ Rel (EulerPaths‘𝐺) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eupth 41365 | . 2 ⊢ EulerPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ 𝑓:(0..^(#‘𝑓))–onto→dom (iEdg‘𝑔))}) | |
2 | 1 | relmptopab 6781 | 1 ⊢ Rel (EulerPaths‘𝐺) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 Vcvv 3173 class class class wbr 4583 dom cdm 5038 Rel wrel 5043 –onto→wfo 5802 ‘cfv 5804 (class class class)co 6549 0cc0 9815 ..^cfzo 12334 #chash 12979 iEdgciedg 25674 TrailSctrls 40899 EulerPathsceupth 41364 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-eupth 41365 |
This theorem is referenced by: eulerpath 41409 |
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