Theorem lo1res 15525

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) → (𝐹 ↾ 𝐴) ∈ ≤𝑂(1))

Proof of Theorem lo1res

Dummy variables 𝑥 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lo1f 15484	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
2		lo1bdd 15486	. . . 4 ⊢ ((𝐹 ∈ ≤𝑂(1) ∧ 𝐹:dom 𝐹⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))
3	1, 2	mpdan 687	. . 3 ⊢ (𝐹 ∈ ≤𝑂(1) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))
4		inss1 4200	. . . . . . 7 ⊢ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹
5		ssralv 4015	. . . . . . 7 ⊢ ((dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹 → (∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)))
6	4, 5	ax-mp 5	. . . . . 6 ⊢ (∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))
7		elinel2 4165	. . . . . . . . . 10 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → 𝑦 ∈ 𝐴)
8	7	fvresd 6878	. . . . . . . . 9 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
9	8	breq1d 5117	. . . . . . . 8 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚))
10	9	imbi2d 340	. . . . . . 7 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚) ↔ (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)))
11	10	ralbiia 3073	. . . . . 6 ⊢ (∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚) ↔ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))
12	6, 11	sylibr 234	. . . . 5 ⊢ (∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚))
13	12	reximi 3067	. . . 4 ⊢ (∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚))
14	13	reximi 3067	. . 3 ⊢ (∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚))
15	3, 14	syl 17	. 2 ⊢ (𝐹 ∈ ≤𝑂(1) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚))
16		fssres 6726	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ)
17	1, 4, 16	sylancl 586	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ)
18		resres 5963	. . . . . 6 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴))
19		ffn 6688	. . . . . . . 8 ⊢ (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹)
20		fnresdm 6637	. . . . . . . 8 ⊢ (𝐹 Fn dom 𝐹 → (𝐹 ↾ dom 𝐹) = 𝐹)
21	1, 19, 20	3syl 18	. . . . . . 7 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ dom 𝐹) = 𝐹)
22	21	reseq1d 5949	. . . . . 6 ⊢ (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴))
23	18, 22	eqtr3id 2778	. . . . 5 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴))
24	23	feq1d 6670	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ ↔ (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ))
25	17, 24	mpbid 232	. . 3 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ)
26		lo1dm 15485	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
27	4, 26	sstrid 3958	. . 3 ⊢ (𝐹 ∈ ≤𝑂(1) → (dom 𝐹 ∩ 𝐴) ⊆ ℝ)
28		ello12 15482	. . 3 ⊢ (((𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ ∧ (dom 𝐹 ∩ 𝐴) ⊆ ℝ) → ((𝐹 ↾ 𝐴) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚)))
29	25, 27, 28	syl2anc 584	. 2 ⊢ (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ 𝐴) ∈ ≤𝑂(1) ↔ ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚)))
30	15, 29	mpbird 257	1 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ 𝐴) ∈ ≤𝑂(1))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3913   ⊆ wss 3914   class class class wbr 5107  dom cdm 5638   ↾ cres 5640   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511  ℝcr 11067   ≤ cle 11209  ≤𝑂(1)clo1 15453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-pre-lttri 11142  ax-pre-lttrn 11143

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-po 5546  df-so 5547  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-er 8671  df-pm 8802  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-ico 13312  df-lo1 15457

This theorem is referenced by:  o1res  15526  lo1res2  15528  lo1resb  15530