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Theorem lo1f 15464
Description: An eventually upper bounded function is a function. (Contributed by Mario Carneiro, 26-May-2016.)
Assertion
Ref Expression
lo1f (𝐹 ∈ ≀𝑂(1) β†’ 𝐹:dom πΉβŸΆβ„)

Proof of Theorem lo1f
Dummy variables π‘₯ π‘š 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ello1 15461 . . 3 (𝐹 ∈ ≀𝑂(1) ↔ (𝐹 ∈ (ℝ ↑pm ℝ) ∧ βˆƒπ‘₯ ∈ ℝ βˆƒπ‘š ∈ ℝ βˆ€π‘¦ ∈ (dom 𝐹 ∩ (π‘₯[,)+∞))(πΉβ€˜π‘¦) ≀ π‘š))
21simplbi 497 . 2 (𝐹 ∈ ≀𝑂(1) β†’ 𝐹 ∈ (ℝ ↑pm ℝ))
3 reex 11198 . . . 4 ℝ ∈ V
43, 3elpm2 8865 . . 3 (𝐹 ∈ (ℝ ↑pm ℝ) ↔ (𝐹:dom πΉβŸΆβ„ ∧ dom 𝐹 βŠ† ℝ))
54simplbi 497 . 2 (𝐹 ∈ (ℝ ↑pm ℝ) β†’ 𝐹:dom πΉβŸΆβ„)
62, 5syl 17 1 (𝐹 ∈ ≀𝑂(1) β†’ 𝐹:dom πΉβŸΆβ„)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062   ∩ cin 3940   βŠ† wss 3941   class class class wbr 5139  dom cdm 5667  βŸΆwf 6530  β€˜cfv 6534  (class class class)co 7402   ↑pm cpm 8818  β„cr 11106  +∞cpnf 11244   ≀ cle 11248  [,)cico 13327  β‰€π‘‚(1)clo1 15433
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 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fv 6542  df-ov 7405  df-oprab 7406  df-mpo 7407  df-pm 8820  df-lo1 15437
This theorem is referenced by:  lo1res  15505  lo1mptrcl  15568
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