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Theorem lo1dm 14868
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 14864 . . 3 (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
21simplbi 500 . 2 (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ))
3 reex 10620 . . . 4 ℝ ∈ V
43, 3elpm2 8430 . . 3 (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
54simprbi 499 . 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 2107  wral 3136  wrex 3137  cin 3933  wss 3934   class class class wbr 5057  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7148  pm cpm 8399  cr 10528  +∞cpnf 10664  cle 10668  [,)cico 12732  ≤𝑂(1)clo1 14836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-pm 8401  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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