Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 14257 | . . 3 ⊢ (𝐹 ∈ 𝑂(1) ↔ (𝐹 ∈ (ℂ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(abs‘(𝐹‘𝑦)) ≤ 𝑚)) | |
2 | 1 | simplbi 476 | . 2 ⊢ (𝐹 ∈ 𝑂(1) → 𝐹 ∈ (ℂ ↑pm ℝ)) |
3 | cnex 10017 | . . . 4 ⊢ ℂ ∈ V | |
4 | reex 10027 | . . . 4 ⊢ ℝ ∈ V | |
5 | 3, 4 | elpm2 7889 | . . 3 ⊢ (𝐹 ∈ (ℂ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℝ)) |
6 | 5 | simprbi 480 | . 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 1990 ∀wral 2912 ∃wrex 2913 ∩ cin 3573 ⊆ wss 3574 class class class wbr 4653 dom cdm 5114 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑pm cpm 7858 ℂcc 9934 ℝcr 9935 +∞cpnf 10071 ≤ cle 10075 [,)cico 12177 abscabs 13974 𝑂(1)co1 14217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-pm 7860 df-o1 14221 |
This theorem is referenced by: o1bdd 14262 lo1o1 14263 o1lo1 14268 o1lo12 14269 o1co 14317 o1of2 14343 o1rlimmul 14349 o1add2 14354 o1mul2 14355 o1sub2 14356 o1dif 14360 o1cxp 24701 |
Copyright terms: Public domain | W3C validator |