Theorem climrel 11290

Description: The limit relation is a relation. (Contributed by NM, 28-Aug-2005.) (Revised by Mario Carneiro, 31-Jan-2014.)

Assertion

Ref	Expression
climrel	|- Rel ~~>

Proof of Theorem climrel

Dummy variables j k x y f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clim 11289	. 2 |- ~~> = { <. f , y >. | ( y e. CC /\ A. x e. RR+ E. j e. ZZ A. k e. ( ZZ>= ` j ) ( ( f ` k ) e. CC /\ ( abs ` ( ( f ` k ) - y ) ) < x ) ) }
2	1	relopabi 4754	1 |- Rel ~~>

Syntax hints:   /\ wa 104   e. wcel 2148   A.wral 2455   E.wrex 2456   class class class wbr 4005   Rel wrel 4633   ` cfv 5218  (class class class)co 5877   CCcc 7811   < clt 7994   - cmin 8130   ZZcz 9255   ZZ>=cuz 9530   RR+crp 9655   abscabs 11008   ~~> cli 11288

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-opab 4067  df-xp 4634  df-rel 4635  df-clim 11289

This theorem is referenced by:  clim  11291  climcl  11292  climi  11297  fclim  11304  climrecl  11334  iserex  11349  climrecvg1n  11358  climcvg1nlem  11359  fsum3cvg3  11406  trirecip  11511  ntrivcvgap0  11559