smflimsuplem3 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem3 45216

Description: The limit of the (𝐻‘𝑛) functions is sigma-measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
smflimsuplem3.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimsuplem3.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smflimsuplem3.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimsuplem3.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem3.e	⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
smflimsuplem3.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))

**Assertion**
Ref	Expression
smflimsuplem3	⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ_≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆))

Distinct variable groups: 𝑥,𝐸 𝑚,𝐹,𝑥 𝑘,𝐻,𝑥 𝑆,𝑘,𝑛 𝑘,𝑍,𝑛,𝑥 𝑚,𝑍,𝑛 𝜑,𝑘,𝑛,𝑥 𝜑,𝑚 Allowed substitution hints: 𝑆(𝑥,𝑚) 𝐸(𝑘,𝑚,𝑛) 𝐹(𝑘,𝑛) 𝐻(𝑚,𝑛) 𝑀(𝑥,𝑘,𝑚,𝑛)

**Proof of Theorem smflimsuplem3**
Step	Hyp	Ref	Expression
1		nfv 1917	. 2 ⊢ Ⅎ𝑛𝜑
2		nfv 1917	. 2 ⊢ Ⅎ𝑥𝜑
3		nfv 1917	. 2 ⊢ Ⅎ𝑘𝜑
4		smflimsuplem3.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		smflimsuplem3.z	. 2 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		fvex 6875	. . . 4 ⊢ (𝐻‘𝑛) ∈ V
7	6	dmex 7868	. . 3 ⊢ dom (𝐻‘𝑛) ∈ V
8	7	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) ∈ V)
9		fvexd 6877	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐻‘𝑛)) → ((𝐻‘𝑛)‘𝑥) ∈ V)
10		smflimsuplem3.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
12		smflimsuplem3.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
13	12	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ}))
14		eqid 2731	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ}
15	5	eluzelz2 43791	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
16		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
17	15, 16	uzn0d 43813	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
18		fvex 6875	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘𝑚) ∈ V
19	18	dmex 7868	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝐹‘𝑚) ∈ V
20	19	rgenw 3064	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
21	20	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
22	17, 21	iinexd 43498	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
23	22	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
24	14, 23	rabexd 5310	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ} ∈ V)
25	13, 24	fvmpt2d 6981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
26		fvres 6881	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) → ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) = (𝐹‘𝑚))
27	26	eqcomd 2737	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
28	27	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
29	28	dmeqd 5881	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → dom (𝐹‘𝑚) = dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
30	29	iineq2dv 4999	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
31	30	eleq2d 2818	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)))
32	27	fveq1d 6864	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) → ((𝐹‘𝑚)‘𝑥) = (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
33	32	mpteq2ia 5228	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
34	33	rneqi 5912	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
35	34	supeq1i 9407	. . . . . . . . . . . . . . 15 ⊢ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < )
36	35	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ))
37	36	eleq1d 2817	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ))
38	31, 37	anbi12d 631	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ) ↔ (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ)))
39	38	rabbidva2 3420	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ})
40	25, 39	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
41	40, 36	mpteq12dv 5216	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^*, < )))
42		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐹 ↾ (ℤ_≥‘𝑛))
43		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹 ↾ (ℤ_≥‘𝑛))
44	15	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
45		smflimsuplem3.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
46	45	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶(SMblFn‘𝑆))
47	5	eleq2i 2824	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑀))
48	47	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑀))
49		uzss 12810	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑀))
50	48, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑀))
51	50, 5	sseqtrrdi 4013	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
52	51	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
53	46, 52	fssresd 6729	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑛)):(ℤ_≥‘𝑛)⟶(SMblFn‘𝑆))
54		eqid 2731	. . . . . . . . . 10 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ}
55		eqid 2731	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ))
56	42, 43, 44, 16, 11, 53, 54, 55	smfsupxr 45210	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥)), ℝ^, < )) ∈ (SMblFn‘𝑆))
57	41, 56	eqeltrd 2832	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) ∈ (SMblFn‘𝑆))
58		smflimsuplem3.h	. . . . . . . 8 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
59	57, 58	fmptd 7082	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑍⟶(SMblFn‘𝑆))
60	59	ffvelcdmda 7055	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ (SMblFn‘𝑆))
61		eqid 2731	. . . . . 6 ⊢ dom (𝐻‘𝑛) = dom (𝐻‘𝑛)
62	11, 60, 61	smff 45126	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):dom (𝐻‘𝑛)⟶ℝ)
63	62	feqmptd 6930	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)))
64	63	eqcomd 2737	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) = (𝐻‘𝑛))
65	64, 60	eqeltrd 2832	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
66		eqid 2731	. 2 ⊢ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ_≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ_≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ }
67		eqid 2731	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ_≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ_≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥))))
68	1, 2, 3, 4, 5, 8, 9, 10, 65, 66, 67	smflimmpt 45204	1 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ_≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 {crab 3418 Vcvv 3459 ⊆ wss 3928 ∪ ciun 4974 ∩ ciin 4975 ↦ cmpt 5208 dom cdm 5653 ran crn 5654 ↾ cres 5655 ⟶wf 6512 ‘cfv 6516 supcsup 9400 ℝcr 11074 ℝ^*cxr 11212 < clt 11213 ℤcz 12523 ℤ_≥cuz 12787 ⇝ cli 15393 SAlgcsalg 44702 SMblFncsmblfn 45089

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-inf2 9601 ax-cc 10395 ax-ac2 10423 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-iin 4977 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-oadd 8436 df-omul 8437 df-er 8670 df-map 8789 df-pm 8790 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-sup 9402 df-inf 9403 df-oi 9470 df-card 9899 df-acn 9902 df-ac 10076 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-n0 12438 df-z 12524 df-uz 12788 df-q 12898 df-rp 12940 df-ioo 13293 df-ioc 13294 df-ico 13295 df-fl 13722 df-seq 13932 df-exp 13993 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-clim 15397 df-rlim 15398 df-rest 17333 df-topgen 17354 df-top 22295 df-bases 22348 df-salg 44703 df-salgen 44707 df-smblfn 45090

This theorem is referenced by: smflimsuplem8 45221

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