Theorem relwlk 16342

Description: The set (Walks ` G ) of all walks on G is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.)

Assertion

Ref	Expression
relwlk	|- Rel (Walks ` G )

Proof of Theorem relwlk

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

Step	Hyp	Ref	Expression
1		df-wlks 16313	. 2 |- Walks = ( g e. _V |-> { <. f , p >. | ( f e. Word dom (iEdg ` g ) /\ p : ( 0 ... (♯ ` f ) ) --> (Vtx ` g ) /\ A. k e. ( 0..^ (♯ ` f ) )if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( (iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( (iEdg ` g ) ` ( f ` k ) ) ) ) } )
2	1	relmptopab 6256	1 |- Rel (Walks ` G )

Syntax hints:  if-wif 986   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2813   C_ wss 3211   {csn 3689   {cpr 3690   dom cdm 4749   Rel wrel 4754   -->wf 5348   ` cfv 5352  (class class class)co 6050   0cc0 8127   1c1 8128   + caddc 8130   ...cfz 10342  ..^cfzo 10476  ♯chash 11138  Word cword 11224  Vtxcvtx 16007  iEdgciedg 16008  Walkscwlks 16312

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 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fv 5360  df-wlks 16313

This theorem is referenced by:  wlkop  16343