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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 14857 | . 2 ⊢ (𝐹 ⇝𝑟 𝐴 → 𝐹 ∈ (ℂ ↑pm ℝ)) | |
2 | cnex 10618 | . . . 4 ⊢ ℂ ∈ V | |
3 | reex 10628 | . . . 4 ⊢ ℝ ∈ V | |
4 | 2, 3 | elpm2 8438 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
5 | 4 | simprbi 499 | . 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 2114 ⊆ wss 3936 class class class wbr 5066 dom cdm 5555 ⟶wf 6351 (class class class)co 7156 ↑pm cpm 8407 ℂcc 10535 ℝcr 10536 ⇝𝑟 crli 14842 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-pm 8409 df-rlim 14846 |
This theorem is referenced by: rlimcl 14860 rlimi 14870 rlimi2 14871 rlimuni 14907 rlimres 14915 rlimeq 14926 rlimcld2 14935 rlimcn1 14945 rlimcn2 14947 rlimo1 14973 o1rlimmul 14975 rlimneg 15003 rlimsqzlem 15005 rlimno1 15010 rlimcxp 25551 |
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