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Theorem lo1dm 14538
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 14534 . . 3 (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
21simplbi 491 . 2 (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ))
3 reex 10282 . . . 4 ℝ ∈ V
43, 3elpm2 8094 . . 3 (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
54simprbi 490 . 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 2155  wral 3055  wrex 3056  cin 3733  wss 3734   class class class wbr 4811  dom cdm 5279  wf 6066  cfv 6070  (class class class)co 6844  pm cpm 8063  cr 10190  +∞cpnf 10327  cle 10331  [,)cico 12382  ≤𝑂(1)clo1 14506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fv 6078  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-pm 8065  df-lo1 14510
This theorem is referenced by:  lo1bdd  14539  lo1o1  14551  o1lo1  14556  o1lo12  14557  lo1res  14578  lo1eq  14587  lo1add  14645  lo1mul  14646  lo1le  14670
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