Theorem climrel 11243

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 11242	. 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 4737	1 |- Rel ~~>

Syntax hints:   /\ wa 103   e. wcel 2141   A.wral 2448   E.wrex 2449   class class class wbr 3989   Rel wrel 4616   ` cfv 5198  (class class class)co 5853   CCcc 7772   < clt 7954   - cmin 8090   ZZcz 9212   ZZ>=cuz 9487   RR+crp 9610   abscabs 10961   ~~> cli 11241

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617  df-rel 4618  df-clim 11242

This theorem is referenced by:  clim  11244  climcl  11245  climi  11250  fclim  11257  climrecl  11287  iserex  11302  climrecvg1n  11311  climcvg1nlem  11312  fsum3cvg3  11359  trirecip  11464  ntrivcvgap0  11512