Theorem eupa0 26501

Description: There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)

Assertion

Ref	Expression
eupa0	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ∅(𝑉 EulPaths ∅){⟨0, 𝐴⟩})

Proof of Theorem eupa0

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		umgra0 25854	. . 3 ⊢ (𝑉 ∈ 𝑊 → 𝑉 UMGrph ∅)
2	1	adantr 480	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝑉 UMGrph ∅)
3		0nn0 11184	. . . 4 ⊢ 0 ∈ ℕ0
4	3	a1i 11	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 0 ∈ ℕ0)
5		f1o0 6085	. . . 4 ⊢ ∅:∅–1-1-onto→∅
6	5	a1i 11	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ∅:∅–1-1-onto→∅)
7		simpr 476	. . . . . 6 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → 𝐴 ∈ 𝑉)
8		f1osng 6089	. . . . . 6 ⊢ ((0 ∈ ℕ0 ∧ 𝐴 ∈ 𝑉) → {⟨0, 𝐴⟩}:{0}–1-1-onto→{𝐴})
9	3, 7, 8	sylancr 694	. . . . 5 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → {⟨0, 𝐴⟩}:{0}–1-1-onto→{𝐴})
10		f1of 6050	. . . . 5 ⊢ ({⟨0, 𝐴⟩}:{0}–1-1-onto→{𝐴} → {⟨0, 𝐴⟩}:{0}⟶{𝐴})
11	9, 10	syl 17	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → {⟨0, 𝐴⟩}:{0}⟶{𝐴})
12	7	snssd 4281	. . . 4 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → {𝐴} ⊆ 𝑉)
13	11, 12	fssd 5970	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → {⟨0, 𝐴⟩}:{0}⟶𝑉)
14		ral0 4028	. . . 4 ⊢ ∀𝑘 ∈ ∅ (∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)}
15	14	a1i 11	. . 3 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ∀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) = ∅
18	16, 17	syl6eq 2660	. . . . . 6 ⊢ (𝑛 = 0 → (1...𝑛) = ∅)
19		f1oeq2 6041	. . . . . 6 ⊢ ((1...𝑛) = ∅ → (∅:(1...𝑛)–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅))
20	18, 19	syl 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})
24	22, 23	ax-mp 5	. . . . . . 7 ⊢ (0...0) = {0}
25	21, 24	syl6eq 2660	. . . . . 6 ⊢ (𝑛 = 0 → (0...𝑛) = {0})
26	25	feq2d 5944	. . . . 5 ⊢ (𝑛 = 0 → ({⟨0, 𝐴⟩}:(0...𝑛)⟶𝑉 ↔ {⟨0, 𝐴⟩}:{0}⟶𝑉))
27	18	raleqdv 3121	. . . . 5 ⊢ (𝑛 = 0 → (∀𝑘 ∈ (1...𝑛)(∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)} ↔ ∀𝑘 ∈ ∅ (∅‘(∅‘𝑘)) = {({⟨0, 𝐴⟩}‘(𝑘 − 1)), ({⟨0, 𝐴⟩}‘𝑘)}))
28	20, 26, 27	3anbi123d 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)})))
29	28	rspcev 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)}))
30	4, 6, 13, 15, 29	syl13anc 1320	. 2 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ∃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)}))))
33	31, 32	ax-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)})))
34	2, 30, 33	sylanbrc 695	1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐴 ∈ 𝑉) → ∅(𝑉 EulPaths ∅){⟨0, 𝐴⟩})

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)