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Theorem eupths 30229
Description: The Eulerian paths on the graph 𝐺. (Contributed by AV, 18-Feb-2021.) (Revised by AV, 29-Oct-2021.)
Hypothesis
Ref Expression
eupths.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
eupths (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}
Distinct variable group:   𝑓,𝐺,𝑝
Allowed substitution hints:   𝐼(𝑓,𝑝)

Proof of Theorem eupths
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . . . 5 (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2 eupths.i . . . . 5 𝐼 = (iEdg‘𝐺)
31, 2eqtr4di 2793 . . . 4 (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
43dmeqd 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))
64, 5syl 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)𝑝𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
86, 7fvmptopab 7487 1 (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝𝑓:(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  ontowfo 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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