Theorem iseupthf1o 30494

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 30493	. 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼))
3		istrl 29985	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))
4	3	anbi1i 635	. . 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 473	. . 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 465	. . . 4 ⊢ ((Fun ◡𝐹 ∧ 𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼) ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹))
7	6	anbi2i 634	. . 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 300	. 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 6828	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹))
10	9	bicomi 227	. . 3 ⊢ ((𝐹:(0..^(♯‘𝐹))–onto→dom 𝐼 ∧ Fun ◡𝐹) ↔ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼)
11	10	anbi2i 634	. 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 300	1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))–1-1-onto→dom 𝐼))

Syntax hints:   ↔ wb 209   ∧ wa 400   = wceq 1567   class class class wbr 5113  ^◡ccnv 5661  dom cdm 5662  Fun wfun 6531  –onto→wfo 6535  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7411  0cc0 11100  ..^cfzo 13682  ♯chash 14366  iEdgciedg 29288  Walkscwlks 29887  Trailsctrls 29979  EulerPathsceupth 30489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-wlks 29890  df-trls 29981  df-eupth 30490

This theorem is referenced by:  eupthi  30495  upgriseupth  30499  eupth0  30506  eupthres  30507  eupthp1  30508