Theorem ello1mpt 15430

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 7054	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ)
3		ello1mpt.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
4		ello12 15425	. . 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 1915	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ≤ 𝑧
7		nffvmpt1 6839	. . . . . . 7 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧)
8		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑥 ≤
9		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑥𝑚
10	7, 8, 9	nfbr 5140	. . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚
11	6, 10	nfim 1897	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚)
12		nfv 1915	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)
13		breq2 5097	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑦 ≤ 𝑧 ↔ 𝑦 ≤ 𝑥))
14		fveq2 6828	. . . . . . 7 ⊢ (𝑧 = 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥))
15	14	breq1d 5103	. . . . . 6 ⊢ (𝑧 = 𝑥 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚 ↔ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))
16	13, 15	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚)))
17	11, 12, 16	cbvralw 3275	. . . 4 ⊢ (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚))
18		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
19		eqid 2733	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)
20	19	fvmpt2 6946	. . . . . . . 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 5103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚 ↔ 𝐵 ≤ 𝑚))
23	22	imbi2d 340	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
24	23	ralbidva 3154	. . . 4 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
25	17, 24	bitrid 283	. . 3 ⊢ (𝜑 → (∀𝑧 ∈ 𝐴 (𝑦 ≤ 𝑧 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑧) ≤ 𝑚) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))
26	25	2rexbidv 3198	. 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 279	1 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ ≤𝑂(1) ↔ ∃𝑦 ∈ ℝ ∃𝑚 ∈ ℝ ∀𝑥 ∈ 𝐴 (𝑦 ≤ 𝑥 → 𝐵 ≤ 𝑚)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057   ⊆ wss 3898   class class class wbr 5093   ↦ cmpt 5174  ⟶wf 6482  ‘cfv 6486  ℝcr 11012   ≤ cle 11154  ≤𝑂(1)clo1 15396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-pre-lttri 11087  ax-pre-lttrn 11088

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-er 8628  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-ico 13253  df-lo1 15400

This theorem is referenced by:  ello1mpt2  15431  ello1d  15432  elo1mpt  15443  o1lo1  15446  lo1resb  15473  lo1add  15536  lo1mul  15537  lo1le  15561