Theorem iseupthf1o 30830

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))⟶1-1ontodom 𝐼))

Proof of Theorem iseupthf1o

Step	Hyp	Ref	Expression
1		eupths.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	iseupth 30829	. 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼))
3		istrl 30306	. . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹))
4	3	anbi1i 636	. . 3 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼) ↔ ((𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹) ∧ 𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼))
5		anass 474	. . 3 ⊢ (((𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝐹) ∧ 𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (Fun ◡𝐹 ∧ 𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼)))
6		ancom 466	. . . 4 ⊢ ((Fun ◡𝐹 ∧ 𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼) ↔ (𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼 ∧ Fun ◡𝐹))
7	6	anbi2i 635	. . 3 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (Fun ◡𝐹 ∧ 𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼)) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼 ∧ Fun ◡𝐹)))
8	4, 5, 7	3bitri 300	. 2 ⊢ ((𝐹(Trails‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼 ∧ Fun ◡𝐹)))
9		dff1o3 6850	. . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))⟶1-1ontodom 𝐼 ↔ (𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼 ∧ Fun ◡𝐹))
10	9	bicomi 227	. . 3 ⊢ ((𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼 ∧ Fun ◡𝐹) ↔ 𝐹:(0..^(♯‘𝐹))⟶1-1ontodom 𝐼)
11	10	anbi2i 635	. 2 ⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐹:(0..^(♯‘𝐹))⟶ontodom 𝐼 ∧ Fun ◡𝐹)) ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))⟶1-1ontodom 𝐼))
12	2, 8, 11	3bitri 300	1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))⟶1-1ontodom 𝐼))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   class class class wbr 5106  ^◡ccnv 5654  dom cdm 5655  Fun wfun 6549  ⟶ontowfo 6553  ⟶1-1ontowf1o 6554  ‘cfv 6555  (class class class)co 7440  0cc0 11224  ..^cfzo 13812  ♯chash 14498  iEdgciedg 29602  Walkscwlks 30204  Trailsctrls 30300  EulerPathsceupth 30825

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-wlks 30207  df-trls 30302  df-eupth 30826

This theorem is used by:  eupthi  30831  upgriseupth  30835  eupth0  30842  eupthres  30843  eupthp1  30844