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Theorem rlimpm 13323
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 13312 . . . . 5
2 opabssxp 5079 . . . . 5
31, 2eqsstri 3533 . . . 4
4 dmss 5207 . . . 4
53, 4ax-mp 5 . . 3
6 dmxpss 5443 . . 3
75, 6sstri 3512 . 2
8 rlimrel 13316 . . 3
98releldmi 5244 . 2
107, 9sseldi 3501 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818  A.wral 2807  E.wrex 2808  C_wss 3475   class class class wbr 4452  {copab 4509  X.cxp 5002  domcdm 5004  `cfv 5593  (class class class)co 6296   cpm 7440   cc 9511   cr 9512   clt 9649   cle 9650   cmin 9828   crp 11249   cabs 13067   crli 13308
This theorem is referenced by:  rlimf  13324  rlimss  13325  rlimclim1  13368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rlim 13312
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