Theorem iseupth 28565

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.)

Hypothesis

Ref	Expression
eupths.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
iseupth	⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))

Proof of Theorem iseupth

Dummy variables 𝑓 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eupths.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	eupths 28564	. 2 ⊢ (EulerPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼)}
3		simpl 483	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → 𝑓 = 𝐹)
4		fveq2 6774	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
5	4	oveq2d 7291	. . . 4 ⊢ (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
6	5	adantr 481	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
7		eqidd 2739	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → dom 𝐼 = dom 𝐼)
8	3, 6, 7	foeq123d 6709	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
9		reltrls 28062	. 2 ⊢ Rel (Trails‘𝐺)
10	2, 8, 9	brfvopabrbr 6872	1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   class class class wbr 5074  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-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  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-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fo 6439  df-fv 6441  df-ov 7278  df-trls 28060  df-eupth 28562

This theorem is referenced by:  iseupthf1o  28566  eupthistrl  28575  eucrctshift  28607