iunhoiioolem - Mathbox for Glauco Siliprandi

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Theorem iunhoiioolem 43888

Description: A n-dimensional open interval expressed as the indexed union of half-open intervals. One side of the double inclusion. (Contributed by Glauco Siliprandi, 8-Apr-2021.)

**Hypotheses**
Ref	Expression
iunhoiioolem.K	⊢ Ⅎ𝑘𝜑
iunhoiioolem.x	⊢ (𝜑 → 𝑋 ∈ Fin)
iunhoiioolem.n	⊢ (𝜑 → 𝑋 ≠ ∅)
iunhoiioolem.a	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
iunhoiioolem.b	⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
iunhoiioolem.f	⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
iunhoiioolem.c	⊢ 𝐶 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )

**Assertion**
Ref	Expression
iunhoiioolem	⊢ (𝜑 → 𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))

Distinct variable groups: 𝐶,𝑛 𝑘,𝐹,𝑛 𝑘,𝑋 𝜑,𝑛 Allowed substitution hints: 𝜑(𝑘) 𝐴(𝑘,𝑛) 𝐵(𝑘,𝑛) 𝐶(𝑘) 𝑋(𝑛)

**Proof of Theorem iunhoiioolem**
Step	Hyp	Ref	Expression
1		iunhoiioolem.K	. . . . . 6 ⊢ Ⅎ𝑘𝜑
2		eqid 2737	. . . . . 6 ⊢ (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) = (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
3		iunhoiioolem.f	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
4		ixpf 8601	. . . . . . . . . . 11 ⊢ (𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵) → 𝐹:𝑋⟶∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵))
5	3, 4	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑋⟶∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵))
6		ioossre 12996	. . . . . . . . . . . . 13 ⊢ (𝐴(,)𝐵) ⊆ ℝ
7	6	rgenw 3073	. . . . . . . . . . . 12 ⊢ ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ
8	7	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
9		iunss 4954	. . . . . . . . . . 11 ⊢ (∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ ↔ ∀𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
10	8, 9	sylibr 237	. . . . . . . . . 10 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝑋 (𝐴(,)𝐵) ⊆ ℝ)
11	5, 10	fssd 6563	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ)
12	11	ffvelrnda 6904	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ)
13		iunhoiioolem.a	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
14	12, 13	resubcld 11260	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
15	13	rexrd 10883	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ^*)
16		iunhoiioolem.b	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
17	3	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵))
18		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋)
19		fvixp2 42411	. . . . . . . . . 10 ⊢ ((𝐹 ∈ X𝑘 ∈ 𝑋 (𝐴(,)𝐵) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ (𝐴(,)𝐵))
20	17, 18, 19	syl2anc 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ (𝐴(,)𝐵))
21		ioogtlb 42708	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐹‘𝑘) ∈ (𝐴(,)𝐵)) → 𝐴 < (𝐹‘𝑘))
22	15, 16, 20, 21	syl3anc 1373	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝐴 < (𝐹‘𝑘))
23	13, 12	posdifd 11419	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐴 < (𝐹‘𝑘) ↔ 0 < ((𝐹‘𝑘) − 𝐴)))
24	22, 23	mpbid 235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 0 < ((𝐹‘𝑘) − 𝐴))
25	14, 24	elrpd 12625	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ⁺)
26	1, 2, 25	rnmptssd 42408	. . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ⁺)
27		iunhoiioolem.c	. . . . . 6 ⊢ 𝐶 = inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )
28		ltso 10913	. . . . . . . 8 ⊢ < Or ℝ
29	28	a1i 11	. . . . . . 7 ⊢ (𝜑 → < Or ℝ)
30		iunhoiioolem.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ Fin)
31	2	rnmptfi 42380	. . . . . . . 8 ⊢ (𝑋 ∈ Fin → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin)
32	30, 31	syl 17	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin)
33		iunhoiioolem.n	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ≠ ∅)
34	1, 14, 2, 33	rnmptn0 6107	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ≠ ∅)
35	1, 2, 14	rnmptssd 42408	. . . . . . 7 ⊢ (𝜑 → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)
36		fiinfcl 9117	. . . . . . 7 ⊢ (( < Or ℝ ∧ (ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ≠ ∅ ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
37	29, 32, 34, 35, 36	syl13anc 1374	. . . . . 6 ⊢ (𝜑 → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
38	27, 37	eqeltrid 2842	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
39	26, 38	sseldd 3902	. . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ⁺)
40		rpgtrecnn 42592	. . . 4 ⊢ (𝐶 ∈ ℝ⁺ → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶)
41	39, 40	syl 17	. . 3 ⊢ (𝜑 → ∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶)
42	3	elexd 3428	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ V)
43	42	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 ∈ V)
44	5	ffnd 6546	. . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑋)
45	44	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 Fn 𝑋)
46		nfv 1922	. . . . . . . . . 10 ⊢ Ⅎ𝑘 𝑛 ∈ ℕ
47	1, 46	nfan 1907	. . . . . . . . 9 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑛 ∈ ℕ)
48		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑘(1 / 𝑛)
49		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑘 <
50		nfmpt1 5153	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
51	50	nfrn 5821	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))
52		nfcv 2904	. . . . . . . . . . . 12 ⊢ Ⅎ𝑘ℝ
53	51, 52, 49	nfinf 9098	. . . . . . . . . . 11 ⊢ Ⅎ𝑘inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < )
54	27, 53	nfcxfr 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑘𝐶
55	48, 49, 54	nfbr 5100	. . . . . . . . 9 ⊢ Ⅎ𝑘(1 / 𝑛) < 𝐶
56	47, 55	nfan 1907	. . . . . . . 8 ⊢ Ⅎ𝑘((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶)
57	13	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
58		nnrecre 11872	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ ℕ → (1 / 𝑛) ∈ ℝ)
59	58	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
60	57, 59	readdcld 10862	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ)
61	60	rexrd 10883	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ^*)
62	61	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ^*)
63	16	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
64	63	adantlr 715	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐵 ∈ ℝ^*)
65		ressxr 10877	. . . . . . . . . . . . . 14 ⊢ ℝ ⊆ ℝ^*
66	65	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℝ ⊆ ℝ^*)
67	11, 66	fssd 6563	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑋⟶ℝ^*)
68	67	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹:𝑋⟶ℝ^*)
69	68	ffvelrnda 6904	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ^*)
70	60	adantlr 715	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ∈ ℝ)
71	12	ad4ant14 752	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ℝ)
72	59	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) ∈ ℝ)
73	35, 38	sseldd 3902	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 ∈ ℝ)
74	73	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐶 ∈ ℝ)
75	14	ad4ant14 752	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
76		simplr 769	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < 𝐶)
77	35	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ)
78	32	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin)
79		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑋 → 𝑘 ∈ 𝑋)
80		ovexd 7248	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 ∈ 𝑋 → ((𝐹‘𝑘) − 𝐴) ∈ V)
81	2	elrnmpt1 5827	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑘 ∈ 𝑋 ∧ ((𝐹‘𝑘) − 𝐴) ∈ V) → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
82	79, 80, 81	syl2anc 587	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 ∈ 𝑋 → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
83	82	adantl 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)))
84		infrefilb 11818	. . . . . . . . . . . . . . . 16 ⊢ ((ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ⊆ ℝ ∧ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)) ∈ Fin ∧ ((𝐹‘𝑘) − 𝐴) ∈ ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴))) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ≤ ((𝐹‘𝑘) − 𝐴))
85	77, 78, 83, 84	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → inf(ran (𝑘 ∈ 𝑋 ↦ ((𝐹‘𝑘) − 𝐴)), ℝ, < ) ≤ ((𝐹‘𝑘) − 𝐴))
86	27, 85	eqbrtrid 5088	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → 𝐶 ≤ ((𝐹‘𝑘) − 𝐴))
87	86	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐶 ≤ ((𝐹‘𝑘) − 𝐴))
88	72, 74, 75, 76, 87	ltletrd 10992	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (1 / 𝑛) < ((𝐹‘𝑘) − 𝐴))
89	57	adantlr 715	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → 𝐴 ∈ ℝ)
90	89, 72, 71	ltaddsub2d 11433	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → ((𝐴 + (1 / 𝑛)) < (𝐹‘𝑘) ↔ (1 / 𝑛) < ((𝐹‘𝑘) − 𝐴)))
91	88, 90	mpbird 260	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) < (𝐹‘𝑘))
92	70, 71, 91	ltled 10980	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐴 + (1 / 𝑛)) ≤ (𝐹‘𝑘))
93		iooltub 42723	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ (𝐹‘𝑘) ∈ (𝐴(,)𝐵)) → (𝐹‘𝑘) < 𝐵)
94	15, 16, 20, 93	syl3anc 1373	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) < 𝐵)
95	94	ad4ant14 752	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) < 𝐵)
96	62, 64, 69, 92, 95	elicod 12985	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) ∧ 𝑘 ∈ 𝑋) → (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵))
97	96	ex 416	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → (𝑘 ∈ 𝑋 → (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
98	56, 97	ralrimi 3137	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵))
99	43, 45, 98	3jca 1130	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
100		elixp2 8582	. . . . . 6 ⊢ (𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝐹‘𝑘) ∈ ((𝐴 + (1 / 𝑛))[,)𝐵)))
101	99, 100	sylibr 237	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ (1 / 𝑛) < 𝐶) → 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
102	101	ex 416	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((1 / 𝑛) < 𝐶 → 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵)))
103	102	reximdva 3193	. . 3 ⊢ (𝜑 → (∃𝑛 ∈ ℕ (1 / 𝑛) < 𝐶 → ∃𝑛 ∈ ℕ 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵)))
104	41, 103	mpd 15	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
105		eliun 4908	. 2 ⊢ (𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵) ↔ ∃𝑛 ∈ ℕ 𝐹 ∈ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))
106	104, 105	sylibr 237	1 ⊢ (𝜑 → 𝐹 ∈ ∪ 𝑛 ∈ ℕ X𝑘 ∈ 𝑋 ((𝐴 + (1 / 𝑛))[,)𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 Ⅎwnf 1791 ∈ wcel 2110 ≠ wne 2940 ∀wral 3061 ∃wrex 3062 Vcvv 3408 ⊆ wss 3866 ∅c0 4237 ∪ ciun 4904 class class class wbr 5053 ↦ cmpt 5135 Or wor 5467 ran crn 5552 Fn wfn 6375 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 Xcixp 8578 Fincfn 8626 infcinf 9057 ℝcr 10728 0cc0 10729 1c1 10730 + caddc 10732 ℝ^*cxr 10866 < clt 10867 ≤ cle 10868 − cmin 11062 / cdiv 11489 ℕcn 11830 ℝ⁺crp 12586 (,)cioo 12935 [,)cico 12937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-ixp 8579 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-ioo 12939 df-ico 12941 df-fl 13367

This theorem is referenced by: iunhoiioo 43889

W3C validator