Theorem ello1d 15559

Description: Sufficient condition for 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)
ello1d.3	⊢ (𝜑 → 𝐶 ∈ ℝ)
ello1d.4	⊢ (𝜑 → 𝑀 ∈ ℝ)
ello1d.5	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ≤ 𝑀)

Assertion

Ref	Expression
ello1d	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝜑,𝑥   𝑥,𝑀

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem ello1d

Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ello1d.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ)
2		ello1d.4	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
3		ello1d.5	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝐶 ≤ 𝑥)) → 𝐵 ≤ 𝑀)
4	3	expr 456	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀))
5	4	ralrimiva 3146	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀))
6		breq1 5146	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ≤ 𝑥 ↔ 𝐶 ≤ 𝑥))
7	6	imbi1d 341	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
8	7	ralbidv 3178	. . . 4 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
9		breq2 5147	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝐵 ≤ 𝑚 ↔ 𝐵 ≤ 𝑀))
10	9	imbi2d 340	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)))
11	10	ralbidv 3178	. . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)))
12	8, 11	rspc2ev 3635	. . 3 ⊢ ((𝐶 ∈ ℝ ∧ 𝑀 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)) → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))
13	1, 2, 5, 12	syl3anc 1373	. 2 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚))
14		ello1mpt.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
15		ello1mpt.2	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
16	14, 15	ello1mpt 15557	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
17	13, 16	mpbird 257	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ⊆ wss 3951   class class class wbr 5143   ↦ cmpt 5225  ℝcr 11154   ≤ cle 11296  ≤𝑂(1)clo1 15523

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 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-pre-lttri 11229  ax-pre-lttrn 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-ico 13393  df-lo1 15527

This theorem is referenced by:  elo1d  15572  o1lo12  15574  icco1  15576  lo1const  15657  dirith2  27572  pntrlog2bndlem4  27624  pntrlog2bndlem6  27627