Theorem releupa 26491

Description: The set (𝑉 EulPaths 𝐸) of all Eulerian paths on ⟨𝑉, 𝐸⟩ is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)

Assertion

Ref	Expression
releupa	⊢ Rel (𝑉 EulPaths 𝐸)

Proof of Theorem releupa

Dummy variables 𝑒 𝑓 𝑘 𝑛 𝑝 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eupa 26490	. 2 ⊢ EulPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑣 UMGrph 𝑒 ∧ ∃𝑛 ∈ ℕ0 (𝑓:(1...𝑛)–1-1-onto→dom 𝑒 ∧ 𝑝:(0...𝑛)⟶𝑣 ∧ ∀𝑘 ∈ (1...𝑛)(𝑒‘(𝑓‘𝑘)) = {(𝑝‘(𝑘 − 1)), (𝑝‘𝑘)}))})
2	1	relmpt2opab 7146	1 ⊢ Rel (𝑉 EulPaths 𝐸)

Syntax hints:   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∀wral 2896  ∃wrex 2897  Vcvv 3173  {cpr 4127   class class class wbr 4583  dom cdm 5038  Rel wrel 5043  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   − cmin 10145  ℕ₀cn0 11169  ...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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  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-ral 2901  df-rex 2902  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-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  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-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-eupa 26490

This theorem is referenced by:  eupath  26508