Theorem climrel 11920

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 11919	. 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 4861	1 |- Rel ~~>

Syntax hints:   /\ wa 104   e. wcel 2202   A.wral 2511   E.wrex 2512   class class class wbr 4093   Rel wrel 4736   ` cfv 5333  (class class class)co 6028   CCcc 8090   < clt 8273   - cmin 8409   ZZcz 9540   ZZ>=cuz 9816   RR+crp 9949   abscabs 11637   ~~> cli 11918

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-xp 4737  df-rel 4738  df-clim 11919

This theorem is referenced by:  clim  11921  climcl  11922  climi  11927  fclim  11934  climrecl  11964  iserex  11979  climrecvg1n  11988  climcvg1nlem  11989  fsum3cvg3  12037  trirecip  12142  ntrivcvgap0  12190