Theorem iseupthf1o 30293

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 30292	. 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
3		istrl 29784	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))
4	3	anbi1i 625	. . 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 624	. . 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 6788	. . . 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 624	. 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 1542   class class class wbr 5100  ^◡ccnv 5631  dom cdm 5632  Fun wfun 6494  –onto→wfo 6498  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  0cc0 11038  ..^cfzo 13582  ♯chash 14265  iEdgciedg 29086  Walkscwlks 29686  Trailsctrls 29778  EulerPathsceupth 30288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-wlks 29689  df-trls 29780  df-eupth 30289

This theorem is referenced by:  eupthi  30294  upgriseupth  30298  eupth0  30305  eupthres  30306  eupthp1  30307