Theorem climrel 14223

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 14219	. 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 5245	1 |- Rel ~~>

Syntax hints:   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   CCcc 9934   < clt 10074   - cmin 10266   ZZcz 11377   ZZ>=cuz 11687   RR+crp 11832   abscabs 13974   ~~> cli 14215

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120  df-rel 5121  df-clim 14219

This theorem is referenced by:  clim  14225  climcl  14230  climi  14241  climrlim2  14278  fclim  14284  climrecl  14314  climge0  14315  iserex  14387  caurcvg2  14408  caucvg  14409  iseralt  14415  fsumcvg3  14460  cvgcmpce  14550  climfsum  14552  climcnds  14583  trirecip  14595  ntrivcvgn0  14630  ovoliunlem1  23270  mbflimlem  23434  abelthlem5  24189  emcllem6  24727  lgamgulmlem4  24758  binomcxplemnn0  38548  binomcxplemnotnn0  38555  climf  39854  sumnnodd  39862  climf2  39898  climd  39904  clim2d  39905  climfv  39923  climuzlem  39975  climlimsup  39992  climlimsupcex  40001  climliminflimsupd  40033  climliminf  40038  liminflimsupclim  40039  ioodvbdlimc1lem2  40147  ioodvbdlimc2lem  40149  stirlinglem12  40302  fouriersw  40448