Theorem wlkmex 16301

Description: If there are walks on a graph, the graph is a set. (Contributed by Jim Kingdon, 1-Feb-2026.)

Assertion

Ref	Expression
wlkmex	|- ( W e. (Walks ` G ) -> G e. _V )

Proof of Theorem wlkmex

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

Step	Hyp	Ref	Expression
1		df-wlks 16300	. 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	mptrcl 5759	1 |- ( W e. (Walks ` G ) -> G e. _V )

Syntax hints:   -> wi 4  if-wif 986   /\ w3a 1005   = wceq 1398   e. wcel 2203   A.wral 2520   _Vcvv 2812   C_ wss 3210   {csn 3688   {cpr 3689   {copab 4169   dom cdm 4748   -->wf 5347   ` cfv 5351  (class class class)co 6049   0cc0 8123   1c1 8124   + caddc 8126   ...cfz 10338  ..^cfzo 10472  ♯chash 11133  Word cword 11217  Vtxcvtx 15994  iEdgciedg 15995  Walkscwlks 16299

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 4227  ax-pow 4286  ax-pr 4321

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 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fv 5359  df-wlks 16300

This theorem is referenced by:  wlkv  16308  wlkcompim  16334  wlkeq  16336