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Theorem lo1res 14224
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 14183 . . . 4 (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
2 lo1bdd 14185 . . . 4 ((𝐹 ∈ ≤𝑂(1) ∧ 𝐹:dom 𝐹⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
31, 2mpdan 701 . . 3 (𝐹 ∈ ≤𝑂(1) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
4 inss1 3811 . . . . . . 7 (dom 𝐹𝐴) ⊆ dom 𝐹
5 ssralv 3645 . . . . . . 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 inss2 3812 . . . . . . . . . . 11 (dom 𝐹𝐴) ⊆ 𝐴
87sseli 3579 . . . . . . . . . 10 (𝑦 ∈ (dom 𝐹𝐴) → 𝑦𝐴)
9 fvres 6164 . . . . . . . . . 10 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
108, 9syl 17 . . . . . . . . 9 (𝑦 ∈ (dom 𝐹𝐴) → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
1110breq1d 4623 . . . . . . . 8 (𝑦 ∈ (dom 𝐹𝐴) → (((𝐹𝐴)‘𝑦) ≤ 𝑚 ↔ (𝐹𝑦) ≤ 𝑚))
1211imbi2d 330 . . . . . . 7 (𝑦 ∈ (dom 𝐹𝐴) → ((𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚) ↔ (𝑥𝑦 → (𝐹𝑦) ≤ 𝑚)))
1312ralbiia 2973 . . . . . 6 (∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚) ↔ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚))
146, 13sylibr 224 . . . . 5 (∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
1514reximi 3005 . . . 4 (∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
1615reximi 3005 . . 3 (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥𝑦 → (𝐹𝑦) ≤ 𝑚) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
173, 16syl 17 . 2 (𝐹 ∈ ≤𝑂(1) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚))
18 fssres 6027 . . . . 5 ((𝐹:dom 𝐹⟶ℝ ∧ (dom 𝐹𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ℝ)
191, 4, 18sylancl 693 . . . 4 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ℝ)
20 resres 5368 . . . . . 6 ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹𝐴))
21 ffn 6002 . . . . . . . 8 (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹)
22 fnresdm 5958 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
231, 21, 223syl 18 . . . . . . 7 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ dom 𝐹) = 𝐹)
2423reseq1d 5355 . . . . . 6 (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹𝐴))
2520, 24syl5eqr 2669 . . . . 5 (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹𝐴)) = (𝐹𝐴))
2625feq1d 5987 . . . 4 (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ (dom 𝐹𝐴)):(dom 𝐹𝐴)⟶ℝ ↔ (𝐹𝐴):(dom 𝐹𝐴)⟶ℝ))
2719, 26mpbid 222 . . 3 (𝐹 ∈ ≤𝑂(1) → (𝐹𝐴):(dom 𝐹𝐴)⟶ℝ)
28 lo1dm 14184 . . . 4 (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
294, 28syl5ss 3594 . . 3 (𝐹 ∈ ≤𝑂(1) → (dom 𝐹𝐴) ⊆ ℝ)
30 ello12 14181 . . 3 (((𝐹𝐴):(dom 𝐹𝐴)⟶ℝ ∧ (dom 𝐹𝐴) ⊆ ℝ) → ((𝐹𝐴) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚)))
3127, 29, 30syl2anc 692 . 2 (𝐹 ∈ ≤𝑂(1) → ((𝐹𝐴) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹𝐴)(𝑥𝑦 → ((𝐹𝐴)‘𝑦) ≤ 𝑚)))
3217, 31mpbird 247 1 (𝐹 ∈ ≤𝑂(1) → (𝐹𝐴) ∈ ≤𝑂(1))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3554  wss 3555   class class class wbr 4613  dom cdm 5074  cres 5076   Fn wfn 5842  wf 5843  cfv 5847  cr 9879  cle 10019  ≤𝑂(1)clo1 14152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-pre-lttri 9954  ax-pre-lttrn 9955
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-er 7687  df-pm 7805  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-ico 12123  df-lo1 14156
This theorem is referenced by:  o1res  14225  lo1res2  14227  lo1resb  14229
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