Theorem reltrls 16236

Description: The set (Trails ` G ) of all trails on G is a set of pairs by our definition of a trail, and so is a relation. (Contributed by AV, 29-Oct-2021.)

Assertion

Ref	Expression
reltrls	|- Rel (Trails ` G )

Proof of Theorem reltrls

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

Step	Hyp	Ref	Expression
1		df-trls 16235	. 2 |- Trails = ( g e. _V |-> { <. f , p >. | ( f (Walks ` g ) p /\ Fun `' f ) } )
2	1	relmptopab 6224	1 |- Rel (Trails ` G )

Syntax hints:   /\ wa 104   _Vcvv 2802   class class class wbr 4088   `'ccnv 4724   Rel wrel 4730   Fun wfun 5320   ` cfv 5326  Walkscwlks 16171  Trailsctrls 16234

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fv 5334  df-trls 16235

This theorem is referenced by: (None)