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Mirrors > Home > MPE Home > Th. List > lo1f | Structured version Visualization version GIF version |
Description: An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.) |
Ref | Expression |
---|---|
lo1f | ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ello1 14094 | . . 3 ⊢ (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹‘𝑦) ≤ 𝑚)) | |
2 | 1 | simplbi 475 | . 2 ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ)) |
3 | reex 9906 | . . . 4 ⊢ ℝ ∈ V | |
4 | 3, 3 | elpm2 7775 | . . 3 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ)) |
5 | 4 | simplbi 475 | . 2 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) → 𝐹:dom 𝐹⟶ℝ) |
6 | 2, 5 | syl 17 | 1 ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ∩ cin 3539 ⊆ wss 3540 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ↑pm cpm 7745 ℝcr 9814 +∞cpnf 9950 ≤ cle 9954 [,)cico 12048 ≤𝑂(1)clo1 14066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-pm 7747 df-lo1 14070 |
This theorem is referenced by: lo1res 14138 lo1mptrcl 14200 |
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