Theorem ello1mpt 15230

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 6989	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
3		ello1mpt.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
4		ello12 15225	. . 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 584	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚)))
6		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ≤ 𝑧
7		nffvmpt1 6785	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
8		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥 ≤
9		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥𝑚
10	7, 8, 9	nfbr 5121	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚
11	6, 10	nfim 1899	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚)
12		nfv 1917	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)
13		breq2 5078	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
14		fveq2 6774	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
15	14	breq1d 5084	. . . . . 6 ⊢ (𝑧 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))
16	13, 15	imbi12d 345	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)))
17	11, 12, 16	cbvralw 3373	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))
18		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
19		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
20	19	fvmpt2 6886	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
21	18, 1, 20	syl2anc 584	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵)
22	21	breq1d 5084	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚 ↔ 𝐵 ≤ 𝑚))
23	22	imbi2d 341	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
24	23	ralbidva 3111	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
25	17, 24	bitrid 282	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
26	25	2rexbidv 3229	. 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 278	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887   class class class wbr 5074   ↦ cmpt 5157  ⟶wf 6429  ‘cfv 6433  ℝcr 10870   ≤ cle 11010  ≤𝑂(1)clo1 15196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-er 8498  df-pm 8618  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-ico 13085  df-lo1 15200

This theorem is referenced by:  ello1mpt2  15231  ello1d  15232  elo1mpt  15243  o1lo1  15246  lo1resb  15273  lo1add  15336  lo1mul  15337  lo1le  15363