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Theorem lo1res 14904
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 14863 . . . 4 (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
2 lo1bdd 14865 . . . 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 4202 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
5 ssralv 4030 . . . . . . 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 4170 . . . . . . . . . 10 (𝑦 ∈ (dom 𝐹𝐴) → 𝑦𝐴)
87fvresd 6683 . . . . . . . . 9 (𝑦 ∈ (dom 𝐹𝐴) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
98breq1d 5067 . . . . . . . 8 (𝑦 ∈ (dom 𝐹𝐴) → (((𝐹𝐴)‘𝑦) ≤ 𝑚 ↔ (𝐹𝑦) ≤ 𝑚))
109imbi2d 342 . . . . . . 7 (𝑦 ∈ (dom 𝐹𝐴) → ((𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ 𝑚)))
1110ralbiia 3161 . . . . . 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 3240 . . . 4 (∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
1413reximi 3240 . . 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 6537 . . . . 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 5859 . . . . . 6 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
19 ffn 6507 . . . . . . . 8 (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹)
20 fnresdm 6459 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
211, 19, 203syl 18 . . . . . . 7 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ dom 𝐹) = 𝐹)
2221reseq1d 5845 . . . . . 6 (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
2318, 22syl5eqr 2867 . . . . 5 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
2423feq1d 6492 . . . 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 14864 . . . 4 (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
274, 26sstrid 3975 . . 3 (𝐹 ∈ ≤𝑂(1) → (dom 𝐹𝐴) ⊆ ℝ)
28 ello12 14861 . . 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 3135  wrex 3136  cin 3932  wss 3933   class class class wbr 5057  dom cdm 5548  cres 5550   Fn wfn 6343  wf 6344  cfv 6348  cr 10524  cle 10664  ≤𝑂(1)clo1 14832
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 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-pre-lttri 10599  ax-pre-lttrn 10600
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 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-er 8278  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-ico 12732  df-lo1 14836
This theorem is referenced by:  o1res  14905  lo1res2  14907  lo1resb  14909
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