Theorem relpths 29791

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 29787	. 2 ⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
2	1	relmptopab 7608	1 ⊢ Rel (Paths‘𝐺)

Syntax hints:   ∧ w3a 1086   = wceq 1541  Vcvv 3440   ∩ cin 3900  ∅c0 4285  {cpr 4582   class class class wbr 5098  ^◡ccnv 5623   ↾ cres 5626   “ cima 5627  Rel wrel 5629  Fun wfun 6486  ‘cfv 6492  (class class class)co 7358  0cc0 11026  1c1 11027  ..^cfzo 13570  ♯chash 14253  Trailsctrls 29762  Pathscpths 29783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fv 6500  df-pths 29787

This theorem is referenced by:  iscycl  29864  cyclnspth  29874