Theorem relpths 26616

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 26612	. 2 |- Paths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' ( p |` ( 1..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) ) = (/) ) } )
2	1	relmptopab 6883	1 |- Rel (Paths ` G )

Syntax hints:   /\ w3a 1037   = wceq 1483   _Vcvv 3200   i^i cin 3573   (/)c0 3915   {cpr 4179   class class class wbr 4653   `'ccnv 5113   |` cres 5116   "cima 5117   Rel wrel 5119   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937  ..^cfzo 12465   #chash 13117  Trailsctrls 26587  Pathscpths 26608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fv 5896  df-pths 26612

This theorem is referenced by:  iscycl  26686  cyclnspth  26695