fsupdm - Mathbox for Glauco Siliprandi

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Theorem fsupdm 47200

Description: The domain of the sup function is defined in Proposition 121F (b) of [Fremlin1], p. 38. Note that this definition of the sup function is quite general, as it does not require the original functions to be sigma-measurable, and it could be applied to uncountable sets of functions. The equality proved here is part of the proof of the fourth statement of Proposition 121H in [Fremlin1], p. 39. (Contributed by Glauco Siliprandi, 24-Jan-2025.)

**Hypotheses**
Ref	Expression
fsupdm.1	⊢ Ⅎ𝑛𝜑
fsupdm.2	⊢ Ⅎ𝑥𝜑
fsupdm.3	⊢ Ⅎ𝑚𝜑
fsupdm.4	⊢ Ⅎ𝑥𝐹
fsupdm.5	⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
fsupdm.6	⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}
fsupdm.7	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))

**Assertion**
Ref	Expression
fsupdm	⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))

Distinct variable groups: 𝐷,𝑚 𝑚,𝐹,𝑦 𝑦,𝐻 𝑚,𝑍,𝑛,𝑥,𝑦 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑚,𝑛) 𝐷(𝑥,𝑦,𝑛) 𝐹(𝑥,𝑛) 𝐻(𝑥,𝑚,𝑛)

**Proof of Theorem fsupdm**
Step	Hyp	Ref	Expression
1		fsupdm.6	. . 3 ⊢ 𝐷 = {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}
2		fsupdm.2	. . . 4 ⊢ Ⅎ𝑥𝜑
3		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑥ℕ
4		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑍
5		fsupdm.7	. . . . . . . . 9 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))
6		nfrab1 3421	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}
7	3, 6	nfmpt 5198	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})
8	4, 7	nfmpt 5198	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑛 ∈ 𝑍 ↦ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))
9	5, 8	nfcxfr 2897	. . . . . . . 8 ⊢ Ⅎ𝑥𝐻
10		nfcv 2899	. . . . . . . 8 ⊢ Ⅎ𝑥𝑛
11	9, 10	nffv 6852	. . . . . . 7 ⊢ Ⅎ𝑥(𝐻‘𝑛)
12		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑥𝑚
13	11, 12	nffv 6852	. . . . . 6 ⊢ Ⅎ𝑥((𝐻‘𝑛)‘𝑚)
14	4, 13	nfiin 4981	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
15	3, 14	nfiun 4980	. . . 4 ⊢ Ⅎ𝑥∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
16		fsupdm.3	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝜑
17		nfv 1916	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
18	16, 17	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
19		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑚 𝑦 ∈ ℝ
20	18, 19	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ)
21		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑚∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦
22	20, 21	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑚(((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
23		fsupdm.1	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛𝜑
24		nfii1 4986	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
25	24	nfcri 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)
26	23, 25	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
27		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑛 𝑦 ∈ ℝ
28	26, 27	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ)
29		nfra1 3262	. . . . . . . . . . 11 ⊢ Ⅎ𝑛∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦
30	28, 29	nfan 1901	. . . . . . . . . 10 ⊢ Ⅎ𝑛(((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
31		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑚 ∈ ℕ
32		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑛 𝑦 < 𝑚
33	30, 31, 32	nf3an 1903	. . . . . . . . 9 ⊢ Ⅎ𝑛((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚)
34		vex 3446	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
35	34	a1i 11	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) → 𝑥 ∈ V)
36		simp-4r 784	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
37	36	3ad2antl1 1187	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
38		simpr 484	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
39		eliinid 45470	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
40	37, 38, 39	syl2anc 585	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
41		simp-4l 783	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑛 ∈ 𝑍) → 𝜑)
42	41	3ad2antl1 1187	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝜑)
43		fsupdm.5	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
44	42, 38, 43	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
45	44, 40	ffvelcdmd 7039	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ^*)
46		simpllr 776	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ)
47	46	rexrd 11194	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ^*)
48	47	3ad2antl1 1187	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑦 ∈ ℝ^*)
49		simpl2 1194	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ)
50	49	nnxrd 45636	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℝ^*)
51		simpl1r 1227	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
52		rspa 3227	. . . . . . . . . . . . 13 ⊢ ((∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
53	51, 38, 52	syl2anc 585	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
54		simpl3 1195	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑦 < 𝑚)
55	45, 48, 50, 53, 54	xrlelttrd 13086	. . . . . . . . . . 11 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) < 𝑚)
56	40, 55	rabidd 45514	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})
57		trud 1552	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → ⊤)
58		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ 𝑍)
59		nfcv 2899	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑛𝑍
60		nnex 12163	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
61	60	mptex 7179	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}) ∈ V
62	61	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((⊤ ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}) ∈ V)
63	59, 5, 62	fvmpt2df 45630	. . . . . . . . . . . . . 14 ⊢ ((⊤ ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))
64	57, 58, 63	syl2anc 585	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (𝐻‘𝑛) = (𝑚 ∈ ℕ ↦ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚}))
65		fsupdm.4	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥𝐹
66	65, 10	nffv 6852	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝐹‘𝑛)
67	66	nfdm 5908	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥dom (𝐹‘𝑛)
68		fvex 6855	. . . . . . . . . . . . . . . 16 ⊢ (𝐹‘𝑛) ∈ V
69	68	dmex 7861	. . . . . . . . . . . . . . 15 ⊢ dom (𝐹‘𝑛) ∈ V
70	67, 69	rabexf 45493	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} ∈ V
71	70	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} ∈ V)
72	64, 71	fvmpt2d 6963	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ℕ) → ((𝐻‘𝑛)‘𝑚) = {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})
73	72	eqcomd 2743	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} = ((𝐻‘𝑛)‘𝑚))
74	38, 49, 73	syl2anc 585	. . . . . . . . . 10 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} = ((𝐻‘𝑛)‘𝑚))
75	56, 74	eleqtrd 2839	. . . . . . . . 9 ⊢ ((((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛)‘𝑚))
76	33, 35, 75	eliind2 45489	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) ∧ 𝑚 ∈ ℕ ∧ 𝑦 < 𝑚) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
77		arch 12410	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → ∃𝑚 ∈ ℕ 𝑦 < 𝑚)
78	77	ad2antlr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) → ∃𝑚 ∈ ℕ 𝑦 < 𝑚)
79	22, 76, 78	reximdd 45507	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) ∧ 𝑦 ∈ ℝ) ∧ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) → ∃𝑚 ∈ ℕ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
80	79	rexlimdva2 3141	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛)) → (∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 → ∃𝑚 ∈ ℕ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)))
81	80	3impia 1118	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) → ∃𝑚 ∈ ℕ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
82		eliun 4952	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ↔ ∃𝑚 ∈ ℕ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
83	81, 82	sylibr 234	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∧ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦) → 𝑥 ∈ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
84	2, 15, 83	rabssd 45501	. . 3 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦} ⊆ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
85	1, 84	eqsstrid 3974	. 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
86		nfcv 2899	. . 3 ⊢ Ⅎ𝑚𝐷
87		nfv 1916	. . . . 5 ⊢ Ⅎ𝑥 𝑚 ∈ ℕ
88	2, 87	nfan 1901	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑚 ∈ ℕ)
89		nfrab1 3421	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦}
90	1, 89	nfcxfr 2897	. . . 4 ⊢ Ⅎ𝑥𝐷
91	23, 31	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑛(𝜑 ∧ 𝑚 ∈ ℕ)
92		nfii1 4986	. . . . . . . . 9 ⊢ Ⅎ𝑛∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
93	92	nfcri 2891	. . . . . . . 8 ⊢ Ⅎ𝑛 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)
94	91, 93	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑛((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))
95	34	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ V)
96		eliinid 45470	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛)‘𝑚))
97	96	adantll 715	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ ((𝐻‘𝑛)‘𝑚))
98		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍)
99		simpllr 776	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℕ)
100	98, 99, 72	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑚) = {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})
101	97, 100	eleqtrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚})
102		rabidim1 3423	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} → 𝑥 ∈ dom (𝐹‘𝑛))
103	101, 102	syl 17	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑥 ∈ dom (𝐹‘𝑛))
104	94, 95, 103	eliind2 45489	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛))
105		nnre 12164	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℝ)
106	105	ad2antlr 728	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑚 ∈ ℝ)
107		breq2 5104	. . . . . . . . 9 ⊢ (𝑦 = 𝑚 → (((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ((𝐹‘𝑛)‘𝑥) ≤ 𝑚))
108	107	ralbidv 3161	. . . . . . . 8 ⊢ (𝑦 = 𝑚 → (∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑚))
109	108	adantl 481	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑦 = 𝑚) → (∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦 ↔ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑚))
110		simplll 775	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝜑)
111	43	3adant3 1133	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → (𝐹‘𝑛):dom (𝐹‘𝑛)⟶ℝ^*)
112		simp3 1139	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → 𝑥 ∈ dom (𝐹‘𝑛))
113	111, 112	ffvelcdmd 7039	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑛)) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ^*)
114	110, 98, 103, 113	syl3anc 1374	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ∈ ℝ^*)
115	99	nnxrd 45636	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → 𝑚 ∈ ℝ^*)
116		rabidim2 45461	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑛) ∣ ((𝐹‘𝑛)‘𝑥) < 𝑚} → ((𝐹‘𝑛)‘𝑥) < 𝑚)
117	101, 116	syl 17	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) < 𝑚)
118	114, 115, 117	xrltled 13076	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) ∧ 𝑛 ∈ 𝑍) → ((𝐹‘𝑛)‘𝑥) ≤ 𝑚)
119	94, 118	ralrimia 3237	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑚)
120	106, 109, 119	rspcedvd 3580	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦)
121	104, 120	rabidd 45514	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ {𝑥 ∈ ∩ 𝑛 ∈ 𝑍 dom (𝐹‘𝑛) ∣ ∃𝑦 ∈ ℝ ∀𝑛 ∈ 𝑍 ((𝐹‘𝑛)‘𝑥) ≤ 𝑦})
122	121, 1	eleqtrrdi 2848	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑥 ∈ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚)) → 𝑥 ∈ 𝐷)
123	88, 14, 90, 122	ssdf2 45500	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ⊆ 𝐷)
124	16, 86, 123	iunssdf 45515	. 2 ⊢ (𝜑 → ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚) ⊆ 𝐷)
125	85, 124	eqssd 3953	1 ⊢ (𝜑 → 𝐷 = ∪ 𝑚 ∈ ℕ ∩ 𝑛 ∈ 𝑍 ((𝐻‘𝑛)‘𝑚))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ⊤wtru 1543 Ⅎwnf 1785 ∈ wcel 2114 Ⅎwnfc 2884 ∀wral 3052 ∃wrex 3062 {crab 3401 Vcvv 3442 ∪ ciun 4948 ∩ ciin 4949 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 ⟶wf 6496 ‘cfv 6500 ℝcr 11037 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 ℕcn 12157

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158

This theorem is referenced by: fsupdm2 47201

W3C validator