MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rlimss Structured version   Visualization version   GIF version

Theorem rlimss 14030
Description: Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.)
Assertion
Ref Expression
rlimss (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)

Proof of Theorem rlimss
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))
54simprbi 478 . 2 (𝐹 ∈ (ℂ ↑pm ℝ) → dom 𝐹 ⊆ ℝ)
61, 5syl 17 1 (𝐹𝑟 𝐴 → dom 𝐹 ⊆ ℝ)
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  rlimi2  14042  rlimuni  14078  rlimres  14086  rlimeq  14097  rlimcld2  14106  rlimcn1  14116  rlimcn2  14118  rlimo1  14144  o1rlimmul  14146  rlimneg  14174  rlimsqzlem  14176  rlimno1  14181  rlimcxp  24445
  Copyright terms: Public domain W3C validator