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Theorem lo1mptrcl 14689
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 . . . . 5 (𝜑 → (𝑥𝐴𝐵) ∈ ≤𝑂(1))
2 lo1f 14586 . . . . 5 ((𝑥𝐴𝐵) ∈ ≤𝑂(1) → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℝ)
31, 2syl 17 . . . 4 (𝜑 → (𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℝ)
4 o1add2.1 . . . . . . 7 ((𝜑𝑥𝐴) → 𝐵𝑉)
54ralrimiva 3145 . . . . . 6 (𝜑 → ∀𝑥𝐴 𝐵𝑉)
6 dmmptg 5849 . . . . . 6 (∀𝑥𝐴 𝐵𝑉 → dom (𝑥𝐴𝐵) = 𝐴)
75, 6syl 17 . . . . 5 (𝜑 → dom (𝑥𝐴𝐵) = 𝐴)
87feq2d 6240 . . . 4 (𝜑 → ((𝑥𝐴𝐵):dom (𝑥𝐴𝐵)⟶ℝ ↔ (𝑥𝐴𝐵):𝐴⟶ℝ))
93, 8mpbid 224 . . 3 (𝜑 → (𝑥𝐴𝐵):𝐴⟶ℝ)
10 eqid 2797 . . . 4 (𝑥𝐴𝐵) = (𝑥𝐴𝐵)
1110fmpt 6604 . . 3 (∀𝑥𝐴 𝐵 ∈ ℝ ↔ (𝑥𝐴𝐵):𝐴⟶ℝ)
129, 11sylibr 226 . 2 (𝜑 → ∀𝑥𝐴 𝐵 ∈ ℝ)
1312r19.21bi 3111 1 ((𝜑𝑥𝐴) → 𝐵 ∈ ℝ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3087  cmpt 4920  dom cdm 5310  wf 6095  cr 10221  ≤𝑂(1)clo1 14555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2375  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-pm 8096  df-lo1 14559
This theorem is referenced by:  lo1add  14694  lo1mul  14695  lo1mul2  14696  lo1sub  14698  lo1le  14719
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