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Mirrors > Home > MPE Home > Th. List > o1dm | Structured version Visualization version GIF version |
Description: An eventually bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
o1dm | ⊢ (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elo1 15235 | . . 3 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) | |
2 | 1 | simplbi 498 | . 2 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | cnex 10952 | . . . 4 ⊢ ℂ ∈ V | |
4 | reex 10962 | . . . 4 ⊢ ℝ ∈ V | |
5 | 3, 4 | elpm2 8662 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
6 | 5 | simprbi 497 | . 2 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) → dom 𝐹 ⊆ ℝ) |
7 | 2, 6 | syl 17 | 1 ⊢ (𝐹 ∈ 𝑂(1) → dom 𝐹 ⊆ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 ∩ cin 3886 ⊆ wss 3887 class class class wbr 5074 dom cdm 5589 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ↑pm cpm 8616 ℂcc 10869 ℝcr 10870 +∞cpnf 11006 ≤ cle 11010 [,)cico 13081 abscabs 14945 𝑂(1)co1 15195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-pm 8618 df-o1 15199 |
This theorem is referenced by: o1bdd 15240 lo1o1 15241 o1lo1 15246 o1lo12 15247 o1co 15295 o1of2 15322 o1rlimmul 15328 o1add2 15333 o1mul2 15334 o1sub2 15335 o1dif 15339 o1cxp 26124 |
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