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Mirrors > Home > MPE Home > Th. List > eupths | Structured version Visualization version GIF version |
Description: The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.) |
Ref | Expression |
---|---|
eupths.i | ⊢ 𝐼 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
eupths | ⊢ (EulerPaths‘𝐺) = {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6907 | . . . . 5 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺)) | |
2 | eupths.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 1, 2 | eqtr4di 2793 | . . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼) |
4 | 3 | dmeqd 5919 | . . 3 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼) |
5 | foeq3 6819 | . . 3 ⊢ (dom (iEdg‘𝑔) = dom 𝐼 → (𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔) ↔ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑔 = 𝐺 → (𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔) ↔ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)) |
7 | df-eupth 30227 | . 2 ⊢ EulerPaths = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))}) | |
8 | 6, 7 | fvmptopab 7487 | 1 ⊢ (EulerPaths‘𝐺) = {〈𝑓, 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 {copab 5210 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-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-iota 6516 df-fun 6565 df-fo 6569 df-fv 6571 df-eupth 30227 |
This theorem is referenced by: iseupth 30230 |
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