MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lo1res Structured version   Visualization version   GIF version

Theorem lo1res 15485
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 15444 . . . 4 (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
2 lo1bdd 15446 . . . 4 ((𝐹 ∈ ≤𝑂(1) ∧ 𝐹:dom 𝐹⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
31, 2mpdan 685 . . 3 (𝐹 ∈ ≤𝑂(1) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
4 inss1 4224 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
5 ssralv 4046 . . . . . . 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 4192 . . . . . . . . . 10 (𝑦 ∈ (dom 𝐹𝐴) → 𝑦𝐴)
87fvresd 6898 . . . . . . . . 9 (𝑦 ∈ (dom 𝐹𝐴) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
98breq1d 5151 . . . . . . . 8 (𝑦 ∈ (dom 𝐹𝐴) → (((𝐹𝐴)‘𝑦) ≤ 𝑚 ↔ (𝐹𝑦) ≤ 𝑚))
109imbi2d 340 . . . . . . 7 (𝑦 ∈ (dom 𝐹𝐴) → ((𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ 𝑚)))
1110ralbiia 3090 . . . . . 6 (∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚) ↔ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
126, 11sylibr 233 . . . . 5 (∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
1312reximi 3083 . . . 4 (∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
1413reximi 3083 . . 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 6744 . . . . 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 5986 . . . . . 6 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
19 ffn 6704 . . . . . . . 8 (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹)
20 fnresdm 6656 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
211, 19, 203syl 18 . . . . . . 7 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ dom 𝐹) = 𝐹)
2221reseq1d 5972 . . . . . 6 (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
2318, 22eqtr3id 2785 . . . . 5 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
2423feq1d 6689 . . . 4 (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ℝ ↔ (𝐹𝐴):(dom 𝐹𝐴)⟶ℝ))
2517, 24mpbid 231 . . 3 (𝐹 ∈ ≤𝑂(1) → (𝐹𝐴):(dom 𝐹𝐴)⟶ℝ)
26 lo1dm 15445 . . . 4 (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
274, 26sstrid 3989 . . 3 (𝐹 ∈ ≤𝑂(1) → (dom 𝐹𝐴) ⊆ ℝ)
28 ello12 15442 . . 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 256 1 (𝐹 ∈ ≤𝑂(1) → (𝐹𝐴) ∈ ≤𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106  wral 3060  wrex 3069  cin 3943  wss 3944   class class class wbr 5141  dom cdm 5669  cres 5671   Fn wfn 6527  wf 6528  cfv 6532  cr 11091  cle 11231  ≤𝑂(1)clo1 15413
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-pre-lttri 11166  ax-pre-lttrn 11167
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-po 5581  df-so 5582  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-er 8686  df-pm 8806  df-en 8923  df-dom 8924  df-sdom 8925  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-ico 13312  df-lo1 15417
This theorem is referenced by:  o1res  15486  lo1res2  15488  lo1resb  15490
  Copyright terms: Public domain W3C validator