Theorem iseupth 30130

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 30129	. 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 6858	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
5	4	oveq2d 7403	. . . 4 ⊢ (𝑓 = 𝐹 → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
6	5	adantr 480	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (0..^(♯‘𝑓)) = (0..^(♯‘𝐹)))
7		eqidd 2730	. . 3 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → dom 𝐼 = dom 𝐼)
8	3, 6, 7	foeq123d 6793	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
9		reltrls 29622	. 2 ⊢ Rel (Trails‘𝐺)
10	2, 8, 9	brfvopabrbr 6965	1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   class class class wbr 5107  dom cdm 5638  –onto→wfo 6509  ‘cfv 6511  (class class class)co 7387  0cc0 11068  ..^cfzo 13615  ♯chash 14295  iEdgciedg 28924  Trailsctrls 29618  EulerPathsceupth 30126

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fo 6517  df-fv 6519  df-ov 7390  df-trls 29620  df-eupth 30127

This theorem is referenced by:  iseupthf1o  30131  eupthistrl  30140  eucrctshift  30172