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Theorem rlimpm 13302
 Description: Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimpm

Proof of Theorem rlimpm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rlim 13291 . . . . 5
2 opabssxp 5064 . . . . 5
31, 2eqsstri 3519 . . . 4
4 dmss 5192 . . . 4
53, 4ax-mp 5 . . 3
6 dmxpss 5428 . . 3
75, 6sstri 3498 . 2
8 rlimrel 13295 . . 3
98releldmi 5229 . 2
107, 9sseldi 3487 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wcel 1804  wral 2793  wrex 2794   wss 3461   class class class wbr 4437  copab 4494   cxp 4987   cdm 4989  cfv 5578  (class class class)co 6281   cpm 7423  cc 9493  cr 9494   clt 9631   cle 9632   cmin 9810  crp 11229  cabs 13046   crli 13287 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996  df-cnv 4997  df-dm 4999  df-rlim 13291 This theorem is referenced by:  rlimf  13303  rlimss  13304  rlimclim1  13347
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