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Theorem lo1f 14965
Description: An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
lo1f (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)

Proof of Theorem lo1f
Dummy variables 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ello1 14962 . . 3 (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
21simplbi 501 . 2 (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ))
3 reex 10706 . . . 4 ℝ ∈ V
43, 3elpm2 8484 . . 3 (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
54simplbi 501 . 2 (𝐹 ∈ (ℝ ↑pm ℝ) → 𝐹:dom 𝐹⟶ℝ)
62, 5syl 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  cr 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
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