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Theorem lo1res 14906
Description: The restriction of an eventually upper bounded function is eventually upper bounded. (Contributed by Mario Carneiro, 15-Sep-2014.)
Assertion
Ref Expression
lo1res (𝐹 ∈ ≤𝑂(1) → (𝐹𝐴) ∈ ≤𝑂(1))

Proof of Theorem lo1res
Dummy variables 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lo1f 14865 . . . 4 (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
2 lo1bdd 14867 . . . 4 ((𝐹 ∈ ≤𝑂(1) ∧ 𝐹:dom 𝐹⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
31, 2mpdan 683 . . 3 (𝐹 ∈ ≤𝑂(1) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
4 inss1 4204 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
5 ssralv 4032 . . . . . . 7 ((dom 𝐹𝐴) ⊆ dom 𝐹 → (∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚)))
64, 5ax-mp 5 . . . . . 6 (∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
7 elinel2 4172 . . . . . . . . . 10 (𝑦 ∈ (dom 𝐹𝐴) → 𝑦𝐴)
87fvresd 6684 . . . . . . . . 9 (𝑦 ∈ (dom 𝐹𝐴) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
98breq1d 5068 . . . . . . . 8 (𝑦 ∈ (dom 𝐹𝐴) → (((𝐹𝐴)‘𝑦) ≤ 𝑚 ↔ (𝐹𝑦) ≤ 𝑚))
109imbi2d 342 . . . . . . 7 (𝑦 ∈ (dom 𝐹𝐴) → ((𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ 𝑚)))
1110ralbiia 3164 . . . . . 6 (∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚) ↔ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
126, 11sylibr 235 . . . . 5 (∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
1312reximi 3243 . . . 4 (∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
1413reximi 3243 . . 3 (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
153, 14syl 17 . 2 (𝐹 ∈ ≤𝑂(1) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
16 fssres 6538 . . . . 5 ((𝐹:dom 𝐹⟶ℝ ∧ (dom 𝐹𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ℝ)
171, 4, 16sylancl 586 . . . 4 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ℝ)
18 resres 5860 . . . . . 6 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
19 ffn 6508 . . . . . . . 8 (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹)
20 fnresdm 6460 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
211, 19, 203syl 18 . . . . . . 7 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ dom 𝐹) = 𝐹)
2221reseq1d 5846 . . . . . 6 (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
2318, 22syl5eqr 2870 . . . . 5 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
2423feq1d 6493 . . . 4 (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ℝ ↔ (𝐹𝐴):(dom 𝐹𝐴)⟶ℝ))
2517, 24mpbid 233 . . 3 (𝐹 ∈ ≤𝑂(1) → (𝐹𝐴):(dom 𝐹𝐴)⟶ℝ)
26 lo1dm 14866 . . . 4 (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
274, 26sstrid 3977 . . 3 (𝐹 ∈ ≤𝑂(1) → (dom 𝐹𝐴) ⊆ ℝ)
28 ello12 14863 . . 3 (((𝐹𝐴):(dom 𝐹𝐴)⟶ℝ ∧ (dom 𝐹𝐴) ⊆ ℝ) → ((𝐹𝐴) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚)))
2925, 27, 28syl2anc 584 . 2 (𝐹 ∈ ≤𝑂(1) → ((𝐹𝐴) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚)))
3015, 29mpbird 258 1 (𝐹 ∈ ≤𝑂(1) → (𝐹𝐴) ∈ ≤𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207   = wceq 1528  wcel 2105  wral 3138  wrex 3139  cin 3934  wss 3935   class class class wbr 5058  dom cdm 5549  cres 5551   Fn wfn 6344  wf 6345  cfv 6349  cr 10525  cle 10665  ≤𝑂(1)clo1 14834
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-pre-lttri 10600  ax-pre-lttrn 10601
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-po 5468  df-so 5469  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8279  df-pm 8399  df-en 8499  df-dom 8500  df-sdom 8501  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-ico 12734  df-lo1 14838
This theorem is referenced by:  o1res  14907  lo1res2  14909  lo1resb  14911
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