Theorem upbdrech 41561

Description: Choice of an upper bound for a nonempty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
upbdrech.a	⊢ (𝜑 → 𝐴 ≠ ∅)
upbdrech.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
upbdrech.bd	⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
upbdrech.c	⊢ 𝐶 = sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )

Assertion

Ref	Expression
upbdrech	⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑦,𝐵,𝑧   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑥)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem upbdrech

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		upbdrech.c	. . 3 ⊢ 𝐶 = sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < )
2		upbdrech.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
3	2	ralrimiva 3180	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ)
4		nfra1 3217	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ
5		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑥 𝑧 ∈ ℝ
6		simp3 1133	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 = 𝐵)
7		rspa 3204	. . . . . . . . . 10 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ)
8	7	3adant3 1127	. . . . . . . . 9 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝐵 ∈ ℝ)
9	6, 8	eqeltrd 2911	. . . . . . . 8 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵) → 𝑧 ∈ ℝ)
10	9	3exp 1114	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ → (𝑥 ∈ 𝐴 → (𝑧 = 𝐵 → 𝑧 ∈ ℝ)))
11	4, 5, 10	rexlimd 3315	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ ℝ))
12	11	abssdv 4043	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ ℝ → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ ℝ)
13	3, 12	syl 17	. . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ ℝ)
14		upbdrech.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≠ ∅)
15		eqidd 2820	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝐵 = 𝐵)
16	15	rgen 3146	. . . . . . 7 ⊢ ∀𝑥 ∈ 𝐴 𝐵 = 𝐵
17		r19.2z 4438	. . . . . . 7 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐵) → ∃𝑥 ∈ 𝐴 𝐵 = 𝐵)
18	14, 16, 17	sylancl 588	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐵 = 𝐵)
19		nfv 1909	. . . . . . 7 ⊢ Ⅎ𝑥𝜑
20		nfre1 3304	. . . . . . . 8 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 𝑧 = 𝐵
21	20	nfex 2337	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑧∃𝑥 ∈ 𝐴 𝑧 = 𝐵
22		simpr 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
23		elex 3511	. . . . . . . . . . . . 13 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ V)
24	2, 23	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ V)
25		isset 3505	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ V ↔ ∃𝑧 𝑧 = 𝐵)
26	24, 25	sylib 220	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧 𝑧 = 𝐵)
27		rspe 3302	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐴 ∧ ∃𝑧 𝑧 = 𝐵) → ∃𝑥 ∈ 𝐴 ∃𝑧 𝑧 = 𝐵)
28	22, 26, 27	syl2anc 586	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑥 ∈ 𝐴 ∃𝑧 𝑧 = 𝐵)
29		rexcom4 3247	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 𝑧 = 𝐵 ↔ ∃𝑧∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
30	28, 29	sylib 220	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑧∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
31	30	3adant3 1127	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 = 𝐵) → ∃𝑧∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
32	31	3exp 1114	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝐵 = 𝐵 → ∃𝑧∃𝑥 ∈ 𝐴 𝑧 = 𝐵)))
33	19, 21, 32	rexlimd 3315	. . . . . 6 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝐵 = 𝐵 → ∃𝑧∃𝑥 ∈ 𝐴 𝑧 = 𝐵))
34	18, 33	mpd 15	. . . . 5 ⊢ (𝜑 → ∃𝑧∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
35		abn0 4334	. . . . 5 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ≠ ∅ ↔ ∃𝑧∃𝑥 ∈ 𝐴 𝑧 = 𝐵)
36	34, 35	sylibr 236	. . . 4 ⊢ (𝜑 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ≠ ∅)
37		upbdrech.bd	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
38		vex 3496	. . . . . . . . . . . . 13 ⊢ 𝑤 ∈ V
39		eqeq1 2823	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑤 → (𝑧 = 𝐵 ↔ 𝑤 = 𝐵))
40	39	rexbidv 3295	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑤 → (∃𝑥 ∈ 𝐴 𝑧 = 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑤 = 𝐵))
41	38, 40	elab 3665	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
42	41	biimpi 218	. . . . . . . . . . 11 ⊢ (𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
43	42	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}) → ∃𝑥 ∈ 𝐴 𝑤 = 𝐵)
44		nfra1 3217	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
45	19, 44	nfan 1894	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
46	20	nfsab 2810	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}
47	45, 46	nfan 1894	. . . . . . . . . . 11 ⊢ Ⅎ𝑥((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵})
48		nfv 1909	. . . . . . . . . . 11 ⊢ Ⅎ𝑥 𝑤 ≤ 𝑦
49		simp3 1133	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵) → 𝑤 = 𝐵)
50		simp1r 1193	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵) → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦)
51		simp2 1132	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵) → 𝑥 ∈ 𝐴)
52		rspa 3204	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝑦)
53	50, 51, 52	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵) → 𝐵 ≤ 𝑦)
54	49, 53	eqbrtrd 5079	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵) → 𝑤 ≤ 𝑦)
55	54	3exp 1114	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → (𝑥 ∈ 𝐴 → (𝑤 = 𝐵 → 𝑤 ≤ 𝑦)))
56	55	adantr 483	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}) → (𝑥 ∈ 𝐴 → (𝑤 = 𝐵 → 𝑤 ≤ 𝑦)))
57	47, 48, 56	rexlimd 3315	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}) → (∃𝑥 ∈ 𝐴 𝑤 = 𝐵 → 𝑤 ≤ 𝑦))
58	43, 57	mpd 15	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) ∧ 𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}) → 𝑤 ≤ 𝑦)
59	58	ralrimiva 3180	. . . . . . . 8 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∀𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑤 ≤ 𝑦)
60	59	3adant2 1126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦) → ∀𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑤 ≤ 𝑦)
61	60	3exp 1114	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ℝ → (∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∀𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑤 ≤ 𝑦)))
62	61	reximdvai 3270	. . . . 5 ⊢ (𝜑 → (∃𝑦 ∈ ℝ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑤 ≤ 𝑦))
63	37, 62	mpd 15	. . . 4 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑤 ≤ 𝑦)
64		suprcl 11593	. . . 4 ⊢ (({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ ℝ ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑤 ≤ 𝑦) → sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ) ∈ ℝ)
65	13, 36, 63, 64	syl3anc 1366	. . 3 ⊢ (𝜑 → sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ) ∈ ℝ)
66	1, 65	eqeltrid 2915	. 2 ⊢ (𝜑 → 𝐶 ∈ ℝ)
67	13	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ ℝ)
68	30, 35	sylibr 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ≠ ∅)
69	63	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∃𝑦 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑤 ≤ 𝑦)
70		elabrexg 41293	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵})
71	22, 2, 70	syl2anc 586	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵})
72		suprub 11594	. . . . 5 ⊢ ((({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ⊆ ℝ ∧ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵} ≠ ∅ ∧ ∃𝑦 ∈ ℝ ∀𝑤 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}𝑤 ≤ 𝑦) ∧ 𝐵 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}) → 𝐵 ≤ sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ))
73	67, 68, 69, 71, 72	syl31anc 1368	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ sup({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = 𝐵}, ℝ, < ))
74	73, 1	breqtrrdi 5099	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ≤ 𝐶)
75	74	ralrimiva 3180	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶)
76	66, 75	jca 514	1 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ∀𝑥 ∈ 𝐴 𝐵 ≤ 𝐶))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1531  ∃wex 1774   ∈ wcel 2108  {cab 2797   ≠ wne 3014  ∀wral 3136  ∃wrex 3137  Vcvv 3493   ⊆ wss 3934  ∅c0 4289   class class class wbr 5057  supcsup 8896  ℝcr 10528   < clt 10667   ≤ cle 10668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-po 5467  df-so 5468  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-sup 8898  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865

This theorem is referenced by:  upbdrech2  41564