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Mirrors > Home > MPE Home > Th. List > iseupth | Structured version Visualization version GIF version |
Description: The property "〈𝐹, 𝑃〉 is an Eulerian path on the graph 𝐺". An Eulerian path is defined as bijection 𝐹 from the edges to a set 0...(𝑁 − 1) and a function 𝑃:(0...𝑁)⟶𝑉 into the vertices such that for each 0 ≤ 𝑘 < 𝑁, 𝐹(𝑘) is an edge from 𝑃(𝑘) to 𝑃(𝑘 + 1). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 18-Feb-2021.) (Revised by AV, 30-Oct-2021.) |
Ref | Expression |
---|---|
eupths.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
iseupth | ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eupths.i | . . 3 ⊢ 𝐼 = (iEdg‘𝐺) | |
2 | 1 | eupths 30229 | . 2 ⊢ (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)} |
3 | simpl 482 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑓 = 𝐹) | |
4 | fveq2 6907 | . . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹)) | |
5 | 4 | oveq2d 7447 | . . . 4 ⊢ (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹))) |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹))) |
7 | eqidd 2736 | . . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → dom 𝐼 = dom 𝐼) | |
8 | 3, 6, 7 | foeq123d 6842 | . 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)) |
9 | reltrls 29727 | . 2 ⊢ Rel (Trails‘𝐺) | |
10 | 2, 8, 9 | brfvopabrbr 7013 | 1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1537 class class class wbr 5148 dom cdm 5689 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ..^cfzo 13691 ♯chash 14366 iEdgciedg 29029 Trailsctrls 29723 EulerPathsceupth 30226 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-fo 6569 df-fv 6571 df-ov 7434 df-trls 29725 df-eupth 30227 |
This theorem is referenced by: iseupthf1o 30231 eupthistrl 30240 eucrctshift 30272 |
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