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Theorem eupa0 26501
 Description: There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
Assertion
Ref Expression
eupa0 ((𝑉𝑊𝐴𝑉) → ∅(𝑉 EulPaths ∅){⟨0, 𝐴⟩})

Proof of Theorem eupa0
Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 umgra0 25854 . . 3 (𝑉𝑊𝑉 UMGrph ∅)
21adantr 480 . 2 ((𝑉𝑊𝐴𝑉) → 𝑉 UMGrph ∅)
3 0nn0 11184 . . . 4 0 ∈ ℕ0
43a1i 11 . . 3 ((𝑉𝑊𝐴𝑉) → 0 ∈ ℕ0)
5 f1o0 6085 . . . 4 ∅:∅–1-1-onto→∅
65a1i 11 . . 3 ((𝑉𝑊𝐴𝑉) → ∅:∅–1-1-onto→∅)
7 simpr 476 . . . . . 6 ((𝑉𝑊𝐴𝑉) → 𝐴𝑉)
8 f1osng 6089 . . . . . 6 ((0 ∈ ℕ0𝐴𝑉) → {⟨0, 𝐴⟩}:{0}–1-1-onto→{𝐴})
93, 7, 8sylancr 694 . . . . 5 ((𝑉𝑊𝐴𝑉) → {⟨0, 𝐴⟩}:{0}–1-1-onto→{𝐴})
10 f1of 6050 . . . . 5 ({⟨0, 𝐴⟩}:{0}–1-1-onto→{𝐴} → {⟨0, 𝐴⟩}:{0}⟶{𝐴})
119, 10syl 17 . . . 4 ((𝑉𝑊𝐴𝑉) → {⟨0, 𝐴⟩}:{0}⟶{𝐴})
127snssd 4281 . . . 4 ((𝑉𝑊𝐴𝑉) → {𝐴} ⊆ 𝑉)
1311, 12fssd 5970 . . 3 ((𝑉𝑊𝐴𝑉) → {⟨0, 𝐴⟩}:{0}⟶𝑉)
14 ral0 4028 . . . 4 𝑘 ∈ ∅ (∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)}
1514a1i 11 . . 3 ((𝑉𝑊𝐴𝑉) → ∀𝑘 ∈ ∅ (∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)})
16 oveq2 6557 . . . . . . 7 (𝑛 = 0 → (1...𝑛) = (1...0))
17 fz10 12233 . . . . . . 7 (1...0) = ∅
1816, 17syl6eq 2660 . . . . . 6 (𝑛 = 0 → (1...𝑛) = ∅)
19 f1oeq2 6041 . . . . . 6 ((1...𝑛) = ∅ → (∅:(1...𝑛)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
2018, 19syl 17 . . . . 5 (𝑛 = 0 → (∅:(1...𝑛)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
21 oveq2 6557 . . . . . . 7 (𝑛 = 0 → (0...𝑛) = (0...0))
22 0z 11265 . . . . . . . 8 0 ∈ ℤ
23 fzsn 12254 . . . . . . . 8 (0 ∈ ℤ → (0...0) = {0})
2422, 23ax-mp 5 . . . . . . 7 (0...0) = {0}
2521, 24syl6eq 2660 . . . . . 6 (𝑛 = 0 → (0...𝑛) = {0})
2625feq2d 5944 . . . . 5 (𝑛 = 0 → ({⟨0, 𝐴⟩}:(0...𝑛)⟶𝑉 ↔ {⟨0, 𝐴⟩}:{0}⟶𝑉))
2718raleqdv 3121 . . . . 5 (𝑛 = 0 → (∀𝑘 ∈ (1...𝑛)(∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)} ↔ ∀𝑘 ∈ ∅ (∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)}))
2820, 26, 273anbi123d 1391 . . . 4 (𝑛 = 0 → ((∅:(1...𝑛)–1-1-onto→∅ ∧ {⟨0, 𝐴⟩}:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)}) ↔ (∅:∅–1-1-onto→∅ ∧ {⟨0, 𝐴⟩}:{0}⟶𝑉 ∧ ∀𝑘 ∈ ∅ (∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)})))
2928rspcev 3282 . . 3 ((0 ∈ ℕ0 ∧ (∅:∅–1-1-onto→∅ ∧ {⟨0, 𝐴⟩}:{0}⟶𝑉 ∧ ∀𝑘 ∈ ∅ (∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)})) → ∃𝑛 ∈ ℕ0 (∅:(1...𝑛)–1-1-onto→∅ ∧ {⟨0, 𝐴⟩}:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)}))
304, 6, 13, 15, 29syl13anc 1320 . 2 ((𝑉𝑊𝐴𝑉) → ∃𝑛 ∈ ℕ0 (∅:(1...𝑛)–1-1-onto→∅ ∧ {⟨0, 𝐴⟩}:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)}))
31 dm0 5260 . . 3 dom ∅ = ∅
32 iseupa 26492 . . 3 (dom ∅ = ∅ → (∅(𝑉 EulPaths ∅){⟨0, 𝐴⟩} ↔ (𝑉 UMGrph ∅ ∧ ∃𝑛 ∈ ℕ0 (∅:(1...𝑛)–1-1-onto→∅ ∧ {⟨0, 𝐴⟩}:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)}))))
3331, 32ax-mp 5 . 2 (∅(𝑉 EulPaths ∅){⟨0, 𝐴⟩} ↔ (𝑉 UMGrph ∅ ∧ ∃𝑛 ∈ ℕ0 (∅:(1...𝑛)–1-1-onto→∅ ∧ {⟨0, 𝐴⟩}:(0...𝑛)⟶𝑉 ∧ ∀𝑘 ∈ (1...𝑛)(∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)})))
342, 30, 33sylanbrc 695 1 ((𝑉𝑊𝐴𝑉) → ∅(𝑉 EulPaths ∅){⟨0, 𝐴⟩})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  ∅c0 3874  {csn 4125  {cpr 4127  ⟨cop 4131   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   − cmin 10145  ℕ0cn0 11169  ℤcz 11254  ...cfz 12197   UMGrph cumg 25841   EulPaths ceup 26489 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-umgra 25842  df-eupa 26490 This theorem is referenced by: (None)
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