Theorem climrel 11863

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 11862	. 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 4857	1 |- Rel ~~>

Syntax hints:   /\ wa 104   e. wcel 2201   A.wral 2509   E.wrex 2510   class class class wbr 4089   Rel wrel 4732   ` cfv 5328  (class class class)co 6023   CCcc 8035   < clt 8219   - cmin 8355   ZZcz 9484   ZZ>=cuz 9760   RR+crp 9893   abscabs 11580   ~~> cli 11861

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-opab 4152  df-xp 4733  df-rel 4734  df-clim 11862

This theorem is referenced by:  clim  11864  climcl  11865  climi  11870  fclim  11877  climrecl  11907  iserex  11922  climrecvg1n  11931  climcvg1nlem  11932  fsum3cvg3  11980  trirecip  12085  ntrivcvgap0  12133