Theorem ello1mpt 15574

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 7112	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
3		ello1mpt.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
4		ello12 15569	. . 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 595	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚)))
6		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ≤ 𝑧
7		nffvmpt1 6894	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
8		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥 ≤
9		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥𝑚
10	7, 8, 9	nfbr 5159	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚
11	6, 10	nfim 1926	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚)
12		nfv 1944	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)
13		breq2 5114	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
14		fveq2 6883	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
15	14	breq1d 5120	. . . . . 6 ⊢ (𝑧 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))
16	13, 15	imbi12d 347	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)))
17	11, 12, 16	cbvralw 3307	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))
18		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
19		eqid 2763	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
20	19	fvmpt2 7003	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
21	18, 1, 20	syl2anc 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
22	21	breq1d 5120	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚 ↔ 𝐵 ≤ 𝑚))
23	22	imbi2d 343	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
24	23	ralbidva 3186	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
25	17, 24	bitrid 286	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
26	25	2rexbidv 3230	. 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 282	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089   ⊆ wss 3906   class class class wbr 5110   ↦ cmpt 5193  ⟶wf 6534  ‘cfv 6538  ℝcr 11100   ≤ cle 11245  ≤𝑂(1)clo1 15540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-pre-lttri 11175  ax-pre-lttrn 11176

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-po 5571  df-so 5572  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8695  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-ico 13379  df-lo1 15544

This theorem is referenced by:  ello1mpt2  15575  ello1d  15576  elo1mpt  15587  o1lo1  15590  lo1resb  15617  lo1add  15680  lo1mul  15681  lo1le  15705