Theorem iseupthf1o 30056

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 30055	. 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
3		istrl 29554	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))
4	3	anbi1i 622	. . 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 467	. . 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 459	. . . 4 ⊢ ((Fun ◡𝐹 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹))
7	6	anbi2i 621	. . 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 296	. 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 6840	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹))
10	9	bicomi 223	. . 3 ⊢ ((𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹) ↔ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)
11	10	anbi2i 621	. 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 296	1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1533   class class class wbr 5143  ^◡ccnv 5671  dom cdm 5672  Fun wfun 6537  –onto→wfo 6541  –1-1-onto→wf1o 6542  ‘cfv 6543  (class class class)co 7416  0cc0 11138  ..^cfzo 13659  ♯chash 14321  iEdgciedg 28854  Walkscwlks 29454  Trailsctrls 29548  EulerPathsceupth 30051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7419  df-wlks 29457  df-trls 29550  df-eupth 30052

This theorem is referenced by:  eupthi  30057  upgriseupth  30061  eupth0  30068  eupthres  30069  eupthp1  30070