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Theorem lo1dm 14458
Description: An eventually upper bounded function's domain is a subset of the reals. (Contributed by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
lo1dm (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)

Proof of Theorem lo1dm
Dummy variables 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ello1 14454 . . 3 (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
21simplbi 485 . 2 (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ))
3 reex 10229 . . . 4 ℝ ∈ V
43, 3elpm2 8041 . . 3 (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
54simprbi 484 . 2 (𝐹 ∈ (ℝ ↑pm ℝ) → dom 𝐹 ⊆ ℝ)
62, 5syl 17 1 (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  wral 3061  wrex 3062  cin 3722  wss 3723   class class class wbr 4786  dom cdm 5249  wf 6027  cfv 6031  (class class class)co 6793  pm cpm 8010  cr 10137  +∞cpnf 10273  cle 10277  [,)cico 12382  ≤𝑂(1)clo1 14426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-pm 8012  df-lo1 14430
This theorem is referenced by:  lo1bdd  14459  lo1o1  14471  o1lo1  14476  o1lo12  14477  lo1res  14498  lo1eq  14507  lo1add  14565  lo1mul  14566  lo1le  14590
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