Theorem iseupth 30180

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 30179	. 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 6840	. . . . 5 ⊢ (𝑓 = 𝐹 → (♯‘𝑓) = (♯‘𝐹))
5	4	oveq2d 7385	. . . 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 6775	. 2 ⊢ ((𝑓 = 𝐹 ∧ 𝑝 = 𝑃) → (𝑓:(0..^(♯‘𝑓))–onto→dom 𝐼 ↔ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
9		reltrls 29673	. 2 ⊢ Rel (Trails‘𝐺)
10	2, 8, 9	brfvopabrbr 6947	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 5102  dom cdm 5631  –onto→wfo 6497  ‘cfv 6499  (class class class)co 7369  0cc0 11044  ..^cfzo 13591  ♯chash 14271  iEdgciedg 28977  Trailsctrls 29669  EulerPathsceupth 30176

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 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-fo 6505  df-fv 6507  df-ov 7372  df-trls 29671  df-eupth 30177

This theorem is referenced by:  iseupthf1o  30181  eupthistrl  30190  eucrctshift  30222