Theorem iseupthf1o 30226

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
iseupthf1o	⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))

Proof of Theorem iseupthf1o

Step	Hyp	Ref	Expression
1		eupths.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	iseupth 30225	. 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
3		istrl 29717	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))
4	3	anbi1i 624	. . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹) ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
5		anass 468	. . 3 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹) ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (Fun ◡𝐹 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)))
6		ancom 460	. . . 4 ⊢ ((Fun ◡𝐹 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹))
7	6	anbi2i 623	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (Fun ◡𝐹 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼)) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹)))
8	4, 5, 7	3bitri 297	. 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹)))
9		dff1o3 6778	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹))
10	9	bicomi 224	. . 3 ⊢ ((𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹) ↔ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)
11	10	anbi2i 623	. 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹)) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))
12	2, 8, 11	3bitri 297	1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   class class class wbr 5096  ^◡ccnv 5621  dom cdm 5622  Fun wfun 6484  –onto→wfo 6488  –1-1-onto→wf1o 6489  ‘cfv 6490  (class class class)co 7356  0cc0 11024  ..^cfzo 13568  ♯chash 14251  iEdgciedg 29019  Walkscwlks 29619  Trailsctrls 29711  EulerPathsceupth 30221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-wlks 29622  df-trls 29713  df-eupth 30222

This theorem is referenced by:  eupthi  30227  upgriseupth  30231  eupth0  30238  eupthres  30239  eupthp1  30240