Theorem reltrls 16426

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 16425	. 2 |- Trails = ( g e. _V |-> { <. f , p >. | ( f (Walks ` g ) p /\ Fun `' f ) } )
2	1	relmptopab 6258	1 |- Rel (Trails ` G )

Syntax hints:   /\ wa 104   _Vcvv 2815   class class class wbr 4111   `'ccnv 4750   Rel wrel 4756   Fun wfun 5348   ` cfv 5354  Walkscwlks 16361  Trailsctrls 16424

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fv 5362  df-trls 16425

This theorem is referenced by: (None)