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Theorem lo1mptrcl 14981
Description: Reverse closure for an eventually upper bounded function. (Contributed by Mario Carneiro, 26-May-2016.)
Hypotheses
Ref Expression
o1add2.1 ((𝜑𝑥𝐴) → 𝐵𝑉)
lo1mptrcl.3 (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))
Assertion
Ref Expression
lo1mptrcl ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Distinct variable groups:   𝑥,𝐴   𝜑,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem lo1mptrcl
StepHypRef Expression
1 lo1mptrcl.3 . . . 4 (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))
2 lo1f 14878 . . . 4 ((𝑥𝐴𝐵) ∈ ≤𝑂(1) → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℝ)
31, 2syl 17 . . 3 (𝜑 → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℝ)
4 o1add2.1 . . . . . 6 ((𝜑𝑥𝐴) → 𝐵𝑉)
54ralrimiva 3185 . . . . 5 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
6 dmmptg 6099 . . . . 5 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
75, 6syl 17 . . . 4 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
87feq2d 6503 . . 3 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℝ ↔ (𝑥𝐴𝐵):𝐴⟶ℝ))
93, 8mpbid 234 . 2 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
109fvmptelrn 6880 1 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1536  wcel 2113  wral 3141  cmpt 5149  dom cdm 5558  wf 6354  cr 10539  ≤𝑂(1)clo1 14847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-resscn 10597
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-pm 8412  df-lo1 14851
This theorem is referenced by:  lo1add  14986  lo1mul  14987  lo1mul2  14988  lo1sub  14990  lo1le  15011
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