Theorem relpths 27221

Description: The set (Paths‘𝐺) of all paths on 𝐺 is a set of pairs by our definition of a path, and so is a relation. (Contributed by AV, 30-Oct-2021.)

Assertion

Ref	Expression
relpths	⊢ Rel (Paths‘𝐺)

Proof of Theorem relpths

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

Step	Hyp	Ref	Expression
1		df-pths 27217	. 2 ⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
2	1	relmptopab 7211	1 ⊢ Rel (Paths‘𝐺)

Syntax hints:   ∧ w3a 1068   = wceq 1507  Vcvv 3409   ∩ cin 3822  ∅c0 4172  {cpr 4437   class class class wbr 4925  ^◡ccnv 5402   ↾ cres 5405   “ cima 5406  Rel wrel 5408  Fun wfun 6179  ‘cfv 6185  (class class class)co 6974  0cc0 10333  1c1 10334  ..^cfzo 12847  ♯chash 13503  Trailsctrls 27190  Pathscpths 27213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fv 6193  df-pths 27217

This theorem is referenced by:  iscycl  27292  cyclnspth  27301