Theorem lo1res 15575

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 15534	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → 𝐹:dom 𝐹⟶ℝ)
2		lo1bdd 15536	. . . 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 4212	. . . . . . 7 ⊢ (dom 𝐹 ∩ 𝐴) ⊆ dom 𝐹
5		ssralv 4027	. . . . . . 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 4177	. . . . . . . . . 10 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → 𝑦 ∈ 𝐴)
8	7	fvresd 6896	. . . . . . . . 9 ⊢ (𝑦 ∈ (dom 𝐹 ∩ 𝐴) → ((𝐹 ↾ 𝐴)‘𝑦) = (𝐹‘𝑦))
9	8	breq1d 5129	. . . . . . . 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 3080	. . . . . 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 3074	. . . 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 3074	. . 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 6744	. . . . 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 5979	. . . . . 6 ⊢ ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ (dom 𝐹 ∩ 𝐴))
19		ffn 6706	. . . . . . . 8 ⊢ (𝐹:dom 𝐹⟶ℝ → 𝐹 Fn dom 𝐹)
20		fnresdm 6657	. . . . . . . 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 5965	. . . . . 6 ⊢ (𝐹 ∈ ≤𝑂(1) → ((𝐹 ↾ dom 𝐹) ↾ 𝐴) = (𝐹 ↾ 𝐴))
23	18, 22	eqtr3id 2784	. . . . 5 ⊢ (𝐹 ∈ ≤𝑂(1) → (𝐹 ↾ (dom 𝐹 ∩ 𝐴)) = (𝐹 ↾ 𝐴))
24	23	feq1d 6690	. . . 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 15535	. . . 4 ⊢ (𝐹 ∈ ≤𝑂(1) → dom 𝐹 ⊆ ℝ)
27	4, 26	sstrid 3970	. . 3 ⊢ (𝐹 ∈ ≤𝑂(1) → (dom 𝐹 ∩ 𝐴) ⊆ ℝ)
28		ello12 15532	. . 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 2108  ∀wral 3051  ∃wrex 3060   ∩ cin 3925   ⊆ wss 3926   class class class wbr 5119  dom cdm 5654   ↾ cres 5656   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  ℝcr 11128   ≤ cle 11270  ≤𝑂(1)clo1 15503

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-cnex 11185  ax-resscn 11186  ax-pre-lttri 11203  ax-pre-lttrn 11204

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-er 8719  df-pm 8843  df-en 8960  df-dom 8961  df-sdom 8962  df-pnf 11271  df-mnf 11272  df-xr 11273  df-ltxr 11274  df-le 11275  df-ico 13368  df-lo1 15507

This theorem is referenced by:  o1res  15576  lo1res2  15578  lo1resb  15580