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Theorem rlimf 14029
Description: Closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimf (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)

Proof of Theorem rlimf
StepHypRef Expression
1 rlimpm 14028 . 2 (𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))
2 cnex 9874 . . . 4 ℂ ∈ V
3 reex 9884 . . . 4 ℝ ∈ V
42, 3elpm2 7753 . . 3 (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
54simplbi 474 . 2 (𝐹 ∈ (ℂ ↑pm ℝ) → 𝐹:dom 𝐹⟶ℂ)
61, 5syl 17 1 (𝐹𝑟 𝐴𝐹:dom 𝐹⟶ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1976  wss 3539   class class class wbr 4577  dom cdm 5028  wf 5786  (class class class)co 6527  pm cpm 7723  cc 9791  cr 9792  𝑟 crli 14013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-pm 7725  df-rlim 14017
This theorem is referenced by:  rlimcl  14031  rlimi  14041  rlimclim1  14073  rlimres  14086  rlimmptrcl  14135  rlimo1  14144  o1rlimmul  14146  dvfsumrlim2  23544  rlimcxp  24445
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