Theorem releupth 30228

Description: The set (EulerPaths‘𝐺) of all Eulerian paths on 𝐺 is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

Assertion

Ref	Expression
releupth	⊢ Rel (EulerPaths‘𝐺)

Proof of Theorem releupth

Dummy variables 𝑓 𝑔 𝑝 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-eupth 30227	. 2 ⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
2	1	relmptopab 7683	1 ⊢ Rel (EulerPaths‘𝐺)

Syntax hints:   ∧ wa 395  Vcvv 3478   class class class wbr 5148  dom cdm 5689  Rel wrel 5694  –onto→wfo 6561  ‘cfv 6563  (class class class)co 7431  0cc0 11153  ..^cfzo 13691  ♯chash 14366  iEdgciedg 29029  Trailsctrls 29723  EulerPathsceupth 30226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-eupth 30227

This theorem is referenced by:  eulerpath  30270