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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 14864 | . . 3 ⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚)) | |
2 | 1 | simplbi 501 | . 2 ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ)) |
3 | reex 10617 | . . . 4 ⊢ ℝ ∈ V | |
4 | 3, 3 | elpm2 8421 | . . 3 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ)) |
5 | 4 | simprbi 500 | . 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 3106 ∃wrex 3107 ∩ cin 3880 ⊆ wss 3881 class class class wbr 5030 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑pm cpm 8390 ℝcr 10525 +∞cpnf 10661 ≤ cle 10665 [,)cico 12728 ≤𝑂(1)clo1 14836 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-pm 8392 df-lo1 14840 |
This theorem is referenced by: lo1bdd 14869 lo1o1 14881 o1lo1 14886 o1lo12 14887 lo1res 14908 lo1eq 14917 lo1add 14975 lo1mul 14976 lo1le 15000 |
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