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| Mirrors > Home > MPE Home > Th. List > lo1dm | Structured version Visualization version GIF version | ||
| Description: An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.) |
| Ref | Expression |
|---|---|
| lo1dm | ⊢ (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ello1 15428 | . . 3 ⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚)) | |
| 2 | 1 | simplbi 497 | . 2 ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ)) |
| 3 | reex 11103 | . . . 4 ⊢ ℝ ∈ V | |
| 4 | 3, 3 | elpm2 8804 | . . 3 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ)) |
| 5 | 4 | simprbi 496 | . 2 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) → dom 𝐹 ⊆ ℝ) |
| 6 | 2, 5 | syl 17 | 1 ⊢ (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2111 ∀wral 3047 ∃wrex 3056 ∩ cin 3896 ⊆ wss 3897 class class class wbr 5093 dom cdm 5619 ⟶wf 6483 ‘cfv 6487 (class class class)co 7352 ↑pm cpm 8757 ℝcr 11011 +∞cpnf 11149 ≤ cle 11153 [,)cico 13253 ≤𝑂(1)clo1 15400 |
| 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 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11068 ax-resscn 11069 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rab 3396 df-v 3438 df-sbc 3737 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-op 4582 df-uni 4859 df-br 5094 df-opab 5156 df-id 5514 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-iota 6443 df-fun 6489 df-fn 6490 df-f 6491 df-fv 6495 df-ov 7355 df-oprab 7356 df-mpo 7357 df-pm 8759 df-lo1 15404 |
| This theorem is referenced by: lo1bdd 15433 lo1o1 15445 o1lo1 15450 o1lo12 15451 lo1res 15472 lo1eq 15481 lo1add 15540 lo1mul 15541 lo1le 15565 |
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