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Mirrors > Home > MPE Home > Th. List > rlimss | Structured version Visualization version GIF version |
Description: Domain closure of a function with a limit in the complex numbers. (Contributed by Mario Carneiro, 16-Sep-2014.) |
Ref | Expression |
---|---|
rlimss | ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimpm 15495 | . 2 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ)) | |
2 | cnex 11228 | . . . 4 ⊢ ℂ ∈ V | |
3 | reex 11238 | . . . 4 ⊢ ℝ ∈ V | |
4 | 2, 3 | elpm2 8893 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
5 | 4 | simprbi 495 | . 2 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → dom 𝐹 ⊆ ℝ) |
6 | 1, 5 | syl 17 | 1 ⊢ (𝐹 ⇝𝑟 𝐴 → dom 𝐹 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ⊆ wss 3947 class class class wbr 5144 dom cdm 5673 ⟶wf 6540 (class class class)co 7414 ↑pm cpm 8846 ℂcc 11145 ℝcr 11146 ⇝𝑟 crli 15480 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3421 df-v 3465 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-br 5145 df-opab 5207 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-fv 6552 df-ov 7417 df-oprab 7418 df-mpo 7419 df-pm 8848 df-rlim 15484 |
This theorem is referenced by: rlimcl 15498 rlimi 15508 rlimi2 15509 rlimuni 15545 rlimres 15553 rlimeq 15564 rlimcld2 15573 rlimcn1 15583 rlimcn3 15585 rlimo1 15612 o1rlimmul 15614 rlimneg 15644 rlimsqzlem 15646 rlimno1 15651 rlimcxp 26997 |
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