Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 14962 | . . 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 10706 | . . . 4 ⊢ ℝ ∈ V | |
4 | 3, 3 | elpm2 8484 | . . 3 ⊢ (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ)) |
5 | 4 | simplbi 501 | . 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 2114 ∀wral 3053 ∃wrex 3054 ∩ cin 3842 ⊆ wss 3843 class class class wbr 5030 dom cdm 5525 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ↑pm cpm 8438 ℝcr 10614 +∞cpnf 10750 ≤ cle 10754 [,)cico 12823 ≤𝑂(1)clo1 14934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-id 5429 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-fv 6347 df-ov 7173 df-oprab 7174 df-mpo 7175 df-pm 8440 df-lo1 14938 |
This theorem is referenced by: lo1res 15006 lo1mptrcl 15069 |
Copyright terms: Public domain | W3C validator |