Theorem rlimrel 14224

Description: The limit relation is a relation. (Contributed by Mario Carneiro, 24-Sep-2014.)

Assertion

Ref	Expression
rlimrel	|- Rel ~~> r

Proof of Theorem rlimrel

Dummy variables w x y z f are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rlim 14220	. 2 |- ~~> r = { <. f , x >. | ( ( f e. ( CC ^pm RR ) /\ x e. CC ) /\ A. y e. RR+ E. z e. RR A. w e. dom f ( z <_ w -> ( abs ` ( ( f ` w ) - x ) ) < y ) ) }
2	1	relopabi 5245	1 |- Rel ~~> r

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   A.wral 2912   E.wrex 2913   class class class wbr 4653   dom cdm 5114   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   ^pm cpm 7858   CCcc 9934   RRcr 9935   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832   abscabs 13974   ~~> r crli 14216

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-rlim 14220

This theorem is referenced by:  rlim  14226  rlimpm  14231  rlimdm  14282  caucvgrlem2  14405  caucvgr  14406  rlimdmafv  41257