Theorem eupths 28564

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)𝑝 ∧ 𝑓:(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.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . . 5 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
2		eupths.i	. . . . 5 ⊢ 𝐼 = (iEdg‘𝐺)
3	1, 2	eqtr4di 2796	. . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐼)
4	3	dmeqd 5814	. . 3 ⊢ (𝑔 = 𝐺 → dom (iEdg‘𝑔) = dom 𝐼)
5		foeq3 6686	. . 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 28562	. 2 ⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
8	6, 7	fvmptopab 7329	1 ⊢ (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   class class class wbr 5074  {copab 5136  dom cdm 5589  –onto→wfo 6431  ‘cfv 6433  (class class class)co 7275  0cc0 10871  ..^cfzo 13382  ♯chash 14044  iEdgciedg 27367  Trailsctrls 28058  EulerPathsceupth 28561

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fo 6439  df-fv 6441  df-eupth 28562

This theorem is referenced by:  iseupth  28565