Theorem smflimsuplem3 41534

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 1992	. 2 ⊢ Ⅎ𝑛𝜑
2		nfv 1992	. 2 ⊢ Ⅎ𝑥𝜑
3		nfv 1992	. 2 ⊢ Ⅎ𝑘𝜑
4		smflimsuplem3.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		smflimsuplem3.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
6		fvex 6362	. . . 4 ⊢ (𝐻‘𝑛) ∈ V
7	6	dmex 7264	. . 3 ⊢ dom (𝐻‘𝑛) ∈ V
8	7	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) ∈ V)
9		fvexd 6364	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐻‘𝑛)) → ((𝐻‘𝑛)‘𝑥) ∈ V)
10		smflimsuplem3.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
12		nfv 1992	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑛 ∈ 𝑍)
13		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐸‘𝑛)
14		nfrab1 3261	. . . . . . . . . 10 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
15		smflimsuplem3.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
16	15	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}))
17		eqid 2760	. . . . . . . . . . . . 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}
18	5	eluzelz2 40125	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
19		eqid 2760	. . . . . . . . . . . . . . . 16 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
20	18, 19	uzn0d 40150	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
21		fvex 6362	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘𝑚) ∈ V
22	21	dmex 7264	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝐹‘𝑚) ∈ V
23	22	rgenw 3062	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
24	23	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
25	20, 24	iinexd 39817	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
26	25	adantl 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
27	17, 26	rabexd 4965	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
28	16, 27	fvmpt2d 6455	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
29		fvres 6368	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) = (𝐹‘𝑚))
30	29	eqcomd 2766	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))
31	30	adantl 473	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))
32	31	dmeqd 5481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → dom (𝐹‘𝑚) = dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))
33	32	iineq2dv 4695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))
34	33	eleq2d 2825	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)))
35	30	fveq1d 6354	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → ((𝐹‘𝑚)‘𝑥) = (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥))
36	35	mpteq2ia 4892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥))
37	36	rneqi 5507	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥))
38	37	supeq1i 8518	. . . . . . . . . . . . . . 15 ⊢ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < )
39	38	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*, < ))
40	39	eleq1d 2824	. . . . . . . . . . . . 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))
41	34, 40	anbi12d 749	. . . . . . . . . . . 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)))
42	41	rabbidva2 3326	. . . . . . . . . . 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})
43	28, 42	eqtrd 2794	. . . . . . . . . 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})
44	12, 13, 14, 43, 39	mpteq12df 4887	. . . . . . . . 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*, < )))
45		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐹 ↾ (ℤ≥‘𝑛))
46		nfcv 2902	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹 ↾ (ℤ≥‘𝑛))
47	18	adantl 473	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
48		smflimsuplem3.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
49	48	adantr 472	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶(SMblFn‘𝑆))
50	5	eleq2i 2831	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
51	50	biimpi 206	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
52		uzss 11900	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
53	51, 52	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
54	53, 5	syl6sseqr 3793	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
55	54	adantl 473	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ⊆ 𝑍)
56	49, 55	fssresd 6232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑛)):(ℤ≥‘𝑛)⟶(SMblFn‘𝑆))
57		eqid 2760	. . . . . . . . . 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}
58		eqid 2760	. . . . . . . . . 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*, < ))
59	45, 46, 47, 19, 11, 56, 57, 58	smfsupxr 41528	. . . . . . . . 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))
60	44, 59	eqeltrd 2839	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) ∈ (SMblFn‘𝑆))
61		smflimsuplem3.h	. . . . . . . 8 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
62	60, 61	fmptd 6548	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑍⟶(SMblFn‘𝑆))
63	62	ffvelrnda 6522	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ (SMblFn‘𝑆))
64		eqid 2760	. . . . . 6 ⊢ dom (𝐻‘𝑛) = dom (𝐻‘𝑛)
65	11, 63, 64	smff 41447	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):dom (𝐻‘𝑛)⟶ℝ)
66	65	feqmptd 6411	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)))
67	66	eqcomd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) = (𝐻‘𝑛))
68	67, 63	eqeltrd 2839	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
69		eqid 2760	. 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 ⇝ }
70		eqid 2760	. 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))))
71	1, 2, 3, 4, 5, 8, 9, 10, 68, 69, 70	smflimmpt 41522	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))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1072   = wceq 1632   ∈ wcel 2139  ∀wral 3050  {crab 3054  Vcvv 3340   ⊆ wss 3715  ∪ ciun 4672  ∩ ciin 4673   ↦ cmpt 4881  dom cdm 5266  ran crn 5267   ↾ cres 5268  ⟶wf 6045  ‘cfv 6049  supcsup 8511  ℝcr 10127  ℝ^*cxr 10265   < clt 10266  ℤcz 11569  ℤ_≥cuz 11879   ⇝ cli 14414  SAlgcsalg 41031  SMblFncsmblfn 41415

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cc 9449  ax-ac2 9477  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-oadd 7733  df-omul 7734  df-er 7911  df-map 8025  df-pm 8026  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-oi 8580  df-card 8955  df-acn 8958  df-ac 9129  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-3 11272  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-ioo 12372  df-ioc 12373  df-ico 12374  df-fl 12787  df-seq 12996  df-exp 13055  df-cj 14038  df-re 14039  df-im 14040  df-sqrt 14174  df-abs 14175  df-clim 14418  df-rlim 14419  df-rest 16285  df-topgen 16306  df-top 20901  df-bases 20952  df-salg 41032  df-salgen 41036  df-smblfn 41416

This theorem is referenced by:  smflimsuplem8  41539