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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 14860 | . . 3 ⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚)) | |
2 | 1 | simplbi 498 | . 2 ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ)) |
3 | reex 10616 | . . . 4 ⊢ ℝ ∈ V | |
4 | 3, 3 | elpm2 8427 | . . 3 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ)) |
5 | 4 | simprbi 497 | . 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 2105 ∀wral 3135 ∃wrex 3136 ∩ cin 3932 ⊆ wss 3933 class class class wbr 5057 dom cdm 5548 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑pm cpm 8396 ℝcr 10524 +∞cpnf 10660 ≤ cle 10664 [,)cico 12728 ≤𝑂(1)clo1 14832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-pm 8398 df-lo1 14836 |
This theorem is referenced by: lo1bdd 14865 lo1o1 14877 o1lo1 14882 o1lo12 14883 lo1res 14904 lo1eq 14913 lo1add 14971 lo1mul 14972 lo1le 14996 |
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