Theorem relpths 30285

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 30281	. 2 ⊢ Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡(𝑝 ↾ (1..^(♯‘𝑓))) ∧ ((𝑝 “ {0, (♯‘𝑓)}) ∩ (𝑝 “ (1..^(♯‘𝑓)))) = ∅)})
2	1	relmptopab 7663	1 ⊢ Rel (Paths‘𝐺)

This proof depends on syntax axioms:   ∧ w3a 1103   = wceq 1570  Vcvv 3451   ∩ cin 3898  ∅c0 4279  {cpr 4586   class class class wbr 5103  ^◡ccnv 5650   ↾ cres 5653   “ cima 5654  Rel wrel 5656  Fun wfun 6525  ‘cfv 6531  (class class class)co 7412  0cc0 11181  1c1 11182  ..^cfzo 13768  ♯chash 14454  Trailsctrls 30255  Pathscpths 30277

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fv 6539  df-pths 30281

This theorem is used by:  iscycl  30360  cyclnspth  30371