Theorem climrel 11591

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 11590	. 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 4803	1 |- Rel ~~>

Syntax hints:   /\ wa 104   e. wcel 2176   A.wral 2484   E.wrex 2485   class class class wbr 4044   Rel wrel 4680   ` cfv 5271  (class class class)co 5944   CCcc 7923   < clt 8107   - cmin 8243   ZZcz 9372   ZZ>=cuz 9648   RR+crp 9775   abscabs 11308   ~~> cli 11589

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-opab 4106  df-xp 4681  df-rel 4682  df-clim 11590

This theorem is referenced by:  clim  11592  climcl  11593  climi  11598  fclim  11605  climrecl  11635  iserex  11650  climrecvg1n  11659  climcvg1nlem  11660  fsum3cvg3  11707  trirecip  11812  ntrivcvgap0  11860