Theorem lo1res 14910

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 14869	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
2		lo1bdd 14871	. . . 4 ⊢ ((𝐹 ∈ ≤𝑂(1) ∧ 𝐹:dom 𝐹⟶ℝ) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))
3	1, 2	mpdan 685	. . 3 ⊢ (𝐹 ∈ ≤𝑂(1) → ∃𝑥 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))
4		inss1 4204	. . . . . . 7 ⊢ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹
5		ssralv 4032	. . . . . . 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 4172	. . . . . . . . . 10 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → 𝑦 ∈ 𝐴)
8	7	fvresd 6684	. . . . . . . . 9 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
9	8	breq1d 5068	. . . . . . . 8 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → (((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚 ↔ (𝐹‘𝑦) ≤ 𝑚))
10	9	imbi2d 343	. . . . . . 7 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚) ↔ (𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚)))
11	10	ralbiia 3164	. . . . . 6 ⊢ (∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚) ↔ ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚))
12	6, 11	sylibr 236	. . . . 5 ⊢ (∀𝑦 ∈ dom 𝐹(𝑥 ≤ 𝑦 → (𝐹‘𝑦) ≤ 𝑚) → ∀𝑦 ∈ (dom 𝐹 ∩ 𝐴)(𝑥 ≤ 𝑦 → ((𝐹 ↾ 𝐴)‘𝑦) ≤ 𝑚))
13	12	reximi 3243	. . . 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 3243	. . 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 6538	. . . . 5 ⊢ ((𝐹:dom 𝐹⟶ℝ ∧ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ)
17	1, 4, 16	sylancl 588	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ)
18		resres 5860	. . . . . 6 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴))
19		ffn 6508	. . . . . . . 8 ⊢ (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹)
20		fnresdm 6460	. . . . . . . 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 5846	. . . . . 6 ⊢ (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴))
23	18, 22	syl5eqr 2870	. . . . 5 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴))
24	23	feq1d 6493	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ (dom 𝐹 ∩ 𝐴)):(dom 𝐹 ∩ 𝐴)⟶ℝ ↔ (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ))
25	17, 24	mpbid 234	. . 3 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ 𝐴):(dom 𝐹 ∩ 𝐴)⟶ℝ)
26		lo1dm 14870	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
27	4, 26	sstrid 3977	. . 3 ⊢ (𝐹 ∈ ≤𝑂(1) → (dom 𝐹 ∩ 𝐴) ⊆ ℝ)
28		ello12 14867	. . 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 586	. 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 259	1 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ 𝐴) ∈ ≤𝑂(1))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  ∀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  ℝcr 10530   ≤ cle 10670  ≤𝑂(1)clo1 14838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-pre-lttri 10605  ax-pre-lttrn 10606

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  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 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  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 7153  df-oprab 7154  df-mpo 7155  df-er 8283  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-ico 12738  df-lo1 14842

This theorem is referenced by:  o1res  14911  lo1res2  14913  lo1resb  14915