Theorem releupth 29956

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 29955	. 2 ⊢ EulerPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ 𝑓:(0..^(♯‘𝑓))–onto→dom (iEdg‘𝑔))})
2	1	relmptopab 7652	1 ⊢ Rel (EulerPaths‘𝐺)

Syntax hints:   ∧ wa 395  Vcvv 3468   class class class wbr 5141  dom cdm 5669  Rel wrel 5674  –onto→wfo 6534  ‘cfv 6536  (class class class)co 7404  0cc0 11109  ..^cfzo 13630  ♯chash 14292  iEdgciedg 28760  Trailsctrls 29451  EulerPathsceupth 29954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-eupth 29955

This theorem is referenced by:  eulerpath  29998