Theorem ello1mpt 14593

Description: Elementhood in the set of eventually upper bounded functions. (Contributed by Mario Carneiro, 26-May-2016.)

Hypotheses

Ref	Expression
ello1mpt.1	⊢ (𝜑 → 𝐴 ⊆ ℝ)
ello1mpt.2	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)

Assertion

Ref	Expression
ello1mpt	⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))

Distinct variable groups:   𝑥,𝑚,𝑦,𝐴   𝐵,𝑚,𝑦   𝜑,𝑚,𝑥,𝑦

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ello1mpt

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ello1mpt.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
2	1	fmpttd 6611	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
3		ello1mpt.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
4		ello12 14588	. . 3 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚)))
5	2, 3, 4	syl2anc 580	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚)))
6		nfv 2010	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ≤ 𝑧
7		nffvmpt1 6422	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
8		nfcv 2941	. . . . . . 7 ⊢ Ⅎ𝑥 ≤
9		nfcv 2941	. . . . . . 7 ⊢ Ⅎ𝑥𝑚
10	7, 8, 9	nfbr 4890	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚
11	6, 10	nfim 1996	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚)
12		nfv 2010	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)
13		breq2 4847	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
14		fveq2 6411	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
15	14	breq1d 4853	. . . . . 6 ⊢ (𝑧 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))
16	13, 15	imbi12d 336	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)))
17	11, 12, 16	cbvral 3350	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))
18		simpr 478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
19		eqid 2799	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
20	19	fvmpt2 6516	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
21	18, 1, 20	syl2anc 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
22	21	breq1d 4853	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚 ↔ 𝐵 ≤ 𝑚))
23	22	imbi2d 332	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
24	23	ralbidva 3166	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
25	17, 24	syl5bb 275	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
26	25	2rexbidv 3238	. 2 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
27	5, 26	bitrd 271	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ∀wral 3089  ∃wrex 3090   ⊆ wss 3769   class class class wbr 4843   ↦ cmpt 4922  ⟶wf 6097  ‘cfv 6101  ℝcr 10223   ≤ cle 10364  ≤𝑂(1)clo1 14559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-cnex 10280  ax-resscn 10281  ax-pre-lttri 10298  ax-pre-lttrn 10299

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-po 5233  df-so 5234  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-er 7982  df-pm 8098  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-xr 10367  df-ltxr 10368  df-le 10369  df-ico 12430  df-lo1 14563

This theorem is referenced by:  ello1mpt2  14594  ello1d  14595  elo1mpt  14606  o1lo1  14609  lo1resb  14636  lo1add  14698  lo1mul  14699  lo1le  14723