Theorem ello1d 15734

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 462	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀))
5	4	ralrimiva 3158	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀))
6		breq1 5111	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝑦 ≤ 𝑥 ↔ 𝐶 ≤ 𝑥))
7	6	imbi1d 344	. . . . 5 ⊢ (𝑦 = 𝐶 → ((𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
8	7	ralbidv 3189	. . . 4 ⊢ (𝑦 = 𝐶 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
9		breq2 5112	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝐵 ≤ 𝑚 ↔ 𝐵 ≤ 𝑀))
10	9	imbi2d 343	. . . . 5 ⊢ (𝑚 = 𝑀 → ((𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)))
11	10	ralbidv 3189	. . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝐶 ≤ 𝑥 → 𝐵 ≤ 𝑀)))
12	8, 11	rspc2ev 3592	. . 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 1399	. 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 15732	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
17	13, 16	mpbird 260	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   class class class wbr 5108   ↦ cmpt 5192  ℝcr 11238   ≤ cle 11383  ≤𝑂(1)clo1 15698

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-pre-lttri 11313  ax-pre-lttrn 11314

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-po 5566  df-so 5567  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-er 8755  df-pm 8888  df-en 9012  df-dom 9013  df-sdom 9014  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-ico 13522  df-lo1 15702

This theorem is used by:  elo1d  15747  o1lo12  15749  icco1  15751  lo1const  15832  dirith2  27912  pntrlog2bndlem4  27964  pntrlog2bndlem6  27967