MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lo1f Structured version   Visualization version   GIF version

Theorem lo1f 15498
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 15495 . . 3 (𝐹 ∈ ≤𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ (𝑥[,)+∞))(𝐹𝑦) ≤ 𝑚))
21simplbi 496 . 2 (𝐹 ∈ ≤𝑂(1) → 𝐹 ∈ (ℝ ↑pm ℝ))
3 reex 11231 . . . 4 ℝ ∈ V
43, 3elpm2 8893 . . 3 (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom 𝐹⟶ℝ ∧ dom 𝐹 ⊆ ℝ))
54simplbi 496 . 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 2098  wral 3050  wrex 3059  cin 3943  wss 3944   class class class wbr 5149  dom cdm 5678  wf 6545  cfv 6549  (class class class)co 7419  pm cpm 8846  cr 11139  +∞cpnf 11277  cle 11281  [,)cico 13361  ≤𝑂(1)clo1 15467
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-pm 8848  df-lo1 15471
This theorem is referenced by:  lo1res  15539  lo1mptrcl  15602
  Copyright terms: Public domain W3C validator