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Theorem rlimpm 12282
 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 12271 . . . . 5
2 opabssxp 4941 . . . . 5
31, 2eqsstri 3370 . . . 4
4 dmss 5060 . . . 4
53, 4ax-mp 8 . . 3
6 dmxpss 5291 . . 3
75, 6sstri 3349 . 2
8 rlimrel 12275 . . 3
98releldmi 5097 . 2
107, 9sseldi 3338 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725  wral 2697  wrex 2698   wss 3312   class class class wbr 4204  copab 4257   cxp 4867   cdm 4869  cfv 5445  (class class class)co 6072   cpm 7010  cc 8977  cr 8978   clt 9109   cle 9110   cmin 9280  crp 10601  cabs 12027   crli 12267 This theorem is referenced by:  rlimf  12283  rlimss  12284  rlimclim1  12327 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-cnv 4877  df-dm 4879  df-rlim 12271
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