smflimsuplem3 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem3 45837

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 1916	. 2 ⊢ Ⅎ𝑛𝜑
2		nfv 1916	. 2 ⊢ Ⅎ𝑥𝜑
3		nfv 1916	. 2 ⊢ Ⅎ𝑘𝜑
4		smflimsuplem3.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		smflimsuplem3.z	. 2 ⊢ 𝑍 = (ℤ_≥‘𝑀)
6		fvex 6904	. . . 4 ⊢ (𝐻‘𝑛) ∈ V
7	6	dmex 7906	. . 3 ⊢ dom (𝐻‘𝑛) ∈ V
8	7	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) ∈ V)
9		fvexd 6906	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐻‘𝑛)) → ((𝐻‘𝑛)‘𝑥) ∈ V)
10		smflimsuplem3.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	adantr 480	. . . . . 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 44412	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
16		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
17	15, 16	uzn0d 44434	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
18		fvex 6904	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘𝑚) ∈ V
19	18	dmex 7906	. . . . . . . . . . . . . . . . 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 44124	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
23	22	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
24	14, 23	rabexd 5333	. . . . . . . . . . . 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 7011	. . . . . . . . . . 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 6910	. . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
29	28	dmeqd 5905	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → dom (𝐹‘𝑚) = dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
30	29	iineq2dv 5022	. . . . . . . . . . . . . 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 6893	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) → ((𝐹‘𝑚)‘𝑥) = (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
33	32	mpteq2ia 5251	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
34	33	rneqi 5936	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
35	34	supeq1i 9446	. . . . . . . . . . . . . . 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 630	. . . . . . . . . . . 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 3433	. . . . . . . . . . 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 5239	. . . . . . . . 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 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
45		smflimsuplem3.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
46	45	adantr 480	. . . . . . . . . . 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 12850	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑀))
50	48, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑀))
51	50, 5	sseqtrrdi 4033	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
52	51	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
53	46, 52	fssresd 6758	. . . . . . . . . 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 45831	. . . . . . . . 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 7115	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑍⟶(SMblFn‘𝑆))
60	59	ffvelcdmda 7086	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ (SMblFn‘𝑆))
61		eqid 2731	. . . . . 6 ⊢ dom (𝐻‘𝑛) = dom (𝐻‘𝑛)
62	11, 60, 61	smff 45747	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):dom (𝐻‘𝑛)⟶ℝ)
63	62	feqmptd 6960	. . . 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 45825	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ∀wral 3060 {crab 3431 Vcvv 3473 ⊆ wss 3948 ∪ ciun 4997 ∩ ciin 4998 ↦ cmpt 5231 dom cdm 5676 ran crn 5677 ↾ cres 5678 ⟶wf 6539 ‘cfv 6543 supcsup 9439 ℝcr 11113 ℝ^*cxr 11252 < clt 11253 ℤcz 12563 ℤ_≥cuz 12827 ⇝ cli 15433 SAlgcsalg 45323 SMblFncsmblfn 45710

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cc 10434 ax-ac2 10462 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 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 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-oadd 8474 df-omul 8475 df-er 8707 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-sup 9441 df-inf 9442 df-oi 9509 df-card 9938 df-acn 9941 df-ac 10115 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-n0 12478 df-z 12564 df-uz 12828 df-q 12938 df-rp 12980 df-ioo 13333 df-ioc 13334 df-ico 13335 df-fl 13762 df-seq 13972 df-exp 14033 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-clim 15437 df-rlim 15438 df-rest 17373 df-topgen 17394 df-top 22617 df-bases 22670 df-salg 45324 df-salgen 45328 df-smblfn 45711

This theorem is referenced by: smflimsuplem8 45842

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