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Theorem rlimss 14855
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 14853 . 2 (𝐹𝑟 𝐴𝐹 ∈ (ℂ ↑pm ℝ))
2 cnex 10610 . . . 4 ℂ ∈ V
3 reex 10620 . . . 4 ℝ ∈ V
42, 3elpm2 8428 . . 3 (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ))
54simprbi 500 . 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 2115  wss 3919   class class class wbr 5052  dom cdm 5542  wf 6339  (class class class)co 7145  pm cpm 8397  cc 10527  cr 10528  𝑟 crli 14838
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7451  ax-cnex 10585  ax-resscn 10586
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-pw 4523  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-pm 8399  df-rlim 14842
This theorem is referenced by:  rlimcl  14856  rlimi  14866  rlimi2  14867  rlimuni  14903  rlimres  14911  rlimeq  14922  rlimcld2  14931  rlimcn1  14941  rlimcn2  14943  rlimo1  14969  o1rlimmul  14971  rlimneg  14999  rlimsqzlem  15001  rlimno1  15006  rlimcxp  25555
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