Theorem smflim 47349

Description: The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . Notice that every function in the sequence can have a different (partial) domain, and the domain of convergence can be decidedly irregular (Remark 121G of [Fremlin1] p. 39 ). (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
smflim.n	⊢ Ⅎ𝑚𝐹
smflim.x	⊢ Ⅎ𝑥𝐹
smflim.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflim.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smflim.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflim.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflim.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
smflim.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))

Assertion

Ref	Expression
smflim	⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Distinct variable groups:   𝑛,𝐹   𝑆,𝑚,𝑛   𝑚,𝑍,𝑥,𝑛   𝜑,𝑚,𝑛

Allowed substitution hints:   𝜑(𝑥)   𝐷(𝑥,𝑚,𝑛)   𝑆(𝑥)   𝐹(𝑥,𝑚)   𝐺(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑚,𝑛)

Proof of Theorem smflim

Dummy variables 𝑖 𝑗 𝑙 𝑦 𝑘 𝑠 𝑡 𝑤 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1937	. 2 ⊢ Ⅎ𝑎𝜑
2		smflim.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
3		smflim.d	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
4		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥𝑍
5		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥(ℤ≥‘𝑛)
6		smflim.x	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
7		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑚
8	6, 7	nffv 6881	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑚)
9	8	nfdm 5932	. . . . . . . 8 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
10	5, 9	nfiin 4985	. . . . . . 7 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
11	4, 10	nfiun 4984	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
12	11	ssrab2f 45693	. . . . 5 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
13	3, 12	eqsstri 3985	. . . 4 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
14	13	a1i 11	. . 3 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚))
15		uzssz 12874	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
16		smflim.z	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ≥‘𝑀)
17	16	eleq2i 2857	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
18	17	biimpi 219	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
19	15, 18	sselid 3937	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
20		uzid 12868	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
21	19, 20	syl 18	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
22	21	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑛))
23	2	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
24		smflim.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
25	24	ffvelcdmda 7069	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
26		eqid 2765	. . . . . . 7 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
27	23, 25, 26	smfdmss 47305	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ⊆ ∪ 𝑆)
28		smflim.n	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝐹
29		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝑛
30	28, 29	nffv 6881	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝐹‘𝑛)
31	30	nfdm 5932	. . . . . . . 8 ⊢ Ⅎ𝑚dom (𝐹‘𝑛)
32		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑚∪ 𝑆
33	31, 32	nfss 3932	. . . . . . 7 ⊢ Ⅎ𝑚dom (𝐹‘𝑛) ⊆ ∪ 𝑆
34		fveq2 6871	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
35	34	dmeqd 5886	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → dom (𝐹‘𝑚) = dom (𝐹‘𝑛))
36	35	sseq1d 3970	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (dom (𝐹‘𝑚) ⊆ ∪ 𝑆 ↔ dom (𝐹‘𝑛) ⊆ ∪ 𝑆))
37	33, 36	rspce 3573	. . . . . 6 ⊢ ((𝑛 ∈ (ℤ≥‘𝑛) ∧ dom (𝐹‘𝑛) ⊆ ∪ 𝑆) → ∃𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆)
38	22, 27, 37	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆)
39		iinss 5017	. . . . 5 ⊢ (∃𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆)
40	38, 39	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆)
41	40	iunssd 5011	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆)
42	14, 41	sstrd 3949	. 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆)
43		nfv 1937	. . . . 5 ⊢ Ⅎ𝑚𝜑
44		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑚𝑦
45		nfmpt1 5204	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
46		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑚dom ⇝
47	45, 46	nfel 2941	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
48		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑍
49		nfii1 4989	. . . . . . . . 9 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
50	48, 49	nfiun 4984	. . . . . . . 8 ⊢ Ⅎ𝑚∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
51	47, 50	nfrabw 3454	. . . . . . 7 ⊢ Ⅎ𝑚{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
52	3, 51	nfcxfr 2925	. . . . . 6 ⊢ Ⅎ𝑚𝐷
53	44, 52	nfel 2941	. . . . 5 ⊢ Ⅎ𝑚 𝑦 ∈ 𝐷
54	43, 53	nfan 1922	. . . 4 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑦 ∈ 𝐷)
55		nfcv 2927	. . . 4 ⊢ Ⅎ𝑤𝐹
56	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
57	24	ffvelcdmda 7069	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
58		eqid 2765	. . . . . 6 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
59	56, 57, 58	smff 47304	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
60	59	adantlr 727	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
61		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑦∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
62		nfv 1937	. . . . . . 7 ⊢ Ⅎ𝑦(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
63		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
64	8, 63	nffv 6881	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑦)
65	4, 64	nfmpt 5203	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
66	65	nfel1 2943	. . . . . . 7 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝
67		fveq2 6871	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
68	67	mpteq2dv 5199	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
69	68	eleq1d 2850	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ))
70	11, 61, 62, 66, 69	cbvrabw 3452	. . . . . 6 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } = {𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ }
71		nfcv 2927	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑙dom (𝐹‘𝑚)
72		nfcv 2927	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝑙
73	28, 72	nffv 6881	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(𝐹‘𝑙)
74	73	nfdm 5932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚dom (𝐹‘𝑙)
75		fveq2 6871	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑙 → (𝐹‘𝑚) = (𝐹‘𝑙))
76	75	dmeqd 5886	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑙 → dom (𝐹‘𝑚) = dom (𝐹‘𝑙))
77	71, 74, 76	cbviin 4996	. . . . . . . . . . . 12 ⊢ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑙 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑙)
78	77	a1i 11	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑙 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑙))
79		fveq2 6871	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (ℤ≥‘𝑛) = (ℤ≥‘𝑖))
80		eqidd 2766	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑖 ∧ 𝑙 ∈ (ℤ≥‘𝑖)) → dom (𝐹‘𝑙) = dom (𝐹‘𝑙))
81	79, 80	iineq12dv 45682	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → ∩ 𝑙 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑙) = ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙))
82	78, 81	eqtrd 2800	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙))
83	82	cbviunv 4999	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙)
84	83	eleq2i 2857	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑦 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙))
85		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑙𝑍
86		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑙((𝐹‘𝑚)‘𝑦)
87	73, 44	nffv 6881	. . . . . . . . . 10 ⊢ Ⅎ𝑚((𝐹‘𝑙)‘𝑦)
88	75	fveq1d 6873	. . . . . . . . . 10 ⊢ (𝑚 = 𝑙 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑙)‘𝑦))
89	48, 85, 86, 87, 88	cbvmptf 5205	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))
90	89	eleq1i 2856	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ )
91	84, 90	anbi12i 639	. . . . . . 7 ⊢ ((𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ) ↔ (𝑦 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∧ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ ))
92	91	rabbia2 3420	. . . . . 6 ⊢ {𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ } = {𝑦 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∣ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ }
93	3, 70, 92	3eqtri 2792	. . . . 5 ⊢ 𝐷 = {𝑦 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∣ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ }
94		fveq2 6871	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑙)‘𝑦) = ((𝐹‘𝑙)‘𝑤))
95	94	mpteq2dv 5199	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)))
96	95	eleq1d 2850	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ ↔ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) ∈ dom ⇝ ))
97	96	cbvrabv 3427	. . . . . 6 ⊢ {𝑦 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∣ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∣ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) ∈ dom ⇝ }
98		fveq2 6871	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑚 → (𝐹‘𝑙) = (𝐹‘𝑚))
99	98	dmeqd 5886	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑚 → dom (𝐹‘𝑙) = dom (𝐹‘𝑚))
100	74, 71, 99	cbviin 4996	. . . . . . . . . . 11 ⊢ ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) = ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚)
101	100	a1i 11	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝑍 → ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) = ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚))
102	101	iuneq2i 4974	. . . . . . . . 9 ⊢ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) = ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚)
103	102	eleq2i 2857	. . . . . . . 8 ⊢ (𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ↔ 𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚))
104		nfcv 2927	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝑤
105	73, 104	nffv 6881	. . . . . . . . . 10 ⊢ Ⅎ𝑚((𝐹‘𝑙)‘𝑤)
106		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑙((𝐹‘𝑚)‘𝑤)
107	98	fveq1d 6873	. . . . . . . . . 10 ⊢ (𝑙 = 𝑚 → ((𝐹‘𝑙)‘𝑤) = ((𝐹‘𝑚)‘𝑤))
108	85, 48, 105, 106, 107	cbvmptf 5205	. . . . . . . . 9 ⊢ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤))
109	108	eleq1i 2856	. . . . . . . 8 ⊢ ((𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)) ∈ dom ⇝ )
110	103, 109	anbi12i 639	. . . . . . 7 ⊢ ((𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∧ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) ∈ dom ⇝ ) ↔ (𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚) ∧ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)) ∈ dom ⇝ ))
111	110	rabbia2 3420	. . . . . 6 ⊢ {𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∣ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) ∈ dom ⇝ } = {𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)) ∈ dom ⇝ }
112	97, 111	eqtri 2788	. . . . 5 ⊢ {𝑦 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∣ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ } = {𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)) ∈ dom ⇝ }
113	93, 112	eqtri 2788	. . . 4 ⊢ 𝐷 = {𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)) ∈ dom ⇝ }
114		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
115	54, 28, 55, 16, 60, 113, 114	fnlimfvre 46246	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))) ∈ ℝ)
116		smflim.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
117		nfrab1 3437	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
118	3, 117	nfcxfr 2925	. . . . 5 ⊢ Ⅎ𝑥𝐷
119		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦𝐷
120		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
121		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑥 ⇝
122	121, 65	nffv 6881	. . . . 5 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
123	68	fveq2d 6875	. . . . 5 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
124	118, 119, 120, 122, 123	cbvmptf 5205	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
125	116, 124	eqtri 2788	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
126	115, 125	fmptd 7099	. 2 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ)
127		smflim.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
128	127	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑀 ∈ ℤ)
129	2	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
130	24	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆))
131		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑙
132	6, 131	nffv 6881	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑙)
133	132, 63	nffv 6881	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑙)‘𝑦)
134	4, 133	nfmpt 5203	. . . . . 6 ⊢ Ⅎ𝑥(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))
135	121, 134	nffv 6881	. . . . 5 ⊢ Ⅎ𝑥( ⇝ ‘(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)))
136		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑙((𝐹‘𝑚)‘𝑥)
137		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝑥
138	73, 137	nffv 6881	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝐹‘𝑙)‘𝑥)
139	75	fveq1d 6873	. . . . . . . . 9 ⊢ (𝑚 = 𝑙 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑙)‘𝑥))
140	48, 85, 136, 138, 139	cbvmptf 5205	. . . . . . . 8 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑥))
141	140	a1i 11	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑥)))
142		simpl 487	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑙 ∈ 𝑍) → 𝑥 = 𝑦)
143	142	fveq2d 6875	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑙 ∈ 𝑍) → ((𝐹‘𝑙)‘𝑥) = ((𝐹‘𝑙)‘𝑦))
144	143	mpteq2dva 5198	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑥)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)))
145	141, 144	eqtrd 2800	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)))
146	145	fveq2d 6875	. . . . 5 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))))
147	118, 119, 120, 135, 146	cbvmptf 5205	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))))
148	116, 147	eqtri 2788	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))))
149		simpr 489	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
150		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑚 <
151		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑎 + (1 / 𝑗))
152	87, 150, 151	nfbr 5152	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
153	152, 74	nfrabw 3454	. . . . . . 7 ⊢ Ⅎ𝑚{𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))}
154		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑚𝑡
155	154, 74	nfin 4179	. . . . . . 7 ⊢ Ⅎ𝑚(𝑡 ∩ dom (𝐹‘𝑙))
156	153, 155	nfeq 2940	. . . . . 6 ⊢ Ⅎ𝑚{𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))
157		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑚𝑆
158	156, 157	nfrabw 3454	. . . . 5 ⊢ Ⅎ𝑚{𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))}
159		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑘{𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))}
160		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑙{𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
161		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑗{𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
162		nfcv 2927	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦dom (𝐹‘𝑙)
163	132	nfdm 5932	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥dom (𝐹‘𝑙)
164		nfcv 2927	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 <
165		nfcv 2927	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑎 + (1 / 𝑗))
166	133, 164, 165	nfbr 5152	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
167		nfv 1937	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))
168		fveq2 6871	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙)‘𝑦) = ((𝐹‘𝑙)‘𝑥))
169	168	breq1d 5115	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))))
170	162, 163, 166, 167, 169	cbvrabw 3452	. . . . . . . . . . 11 ⊢ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))}
171	170	a1i 11	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))})
172		ineq1 4168	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝑡 ∩ dom (𝐹‘𝑙)) = (𝑠 ∩ dom (𝐹‘𝑙)))
173	171, 172	eqeq12d 2781	. . . . . . . . 9 ⊢ (𝑡 = 𝑠 → ({𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙)) ↔ {𝑥 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑙))))
174	173	cbvrabv 3427	. . . . . . . 8 ⊢ {𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑙))}
175	174	a1i 11	. . . . . . 7 ⊢ (𝑙 = 𝑚 → {𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑙))})
176	99	eleq2d 2851	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑚 → (𝑥 ∈ dom (𝐹‘𝑙) ↔ 𝑥 ∈ dom (𝐹‘𝑚)))
177	98	fveq1d 6873	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑚 → ((𝐹‘𝑙)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
178	177	breq1d 5115	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑚 → (((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))))
179	176, 178	anbi12d 643	. . . . . . . . . 10 ⊢ (𝑙 = 𝑚 → ((𝑥 ∈ dom (𝐹‘𝑙) ∧ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))) ↔ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)))))
180	179	rabbidva2 3419	. . . . . . . . 9 ⊢ (𝑙 = 𝑚 → {𝑥 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))})
181	99	ineq2d 4175	. . . . . . . . 9 ⊢ (𝑙 = 𝑚 → (𝑠 ∩ dom (𝐹‘𝑙)) = (𝑠 ∩ dom (𝐹‘𝑚)))
182	180, 181	eqeq12d 2781	. . . . . . . 8 ⊢ (𝑙 = 𝑚 → ({𝑥 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑙)) ↔ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))))
183	182	rabbidv 3424	. . . . . . 7 ⊢ (𝑙 = 𝑚 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑙))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))})
184	175, 183	eqtrd 2800	. . . . . 6 ⊢ (𝑙 = 𝑚 → {𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))})
185		oveq2 7408	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (1 / 𝑗) = (1 / 𝑘))
186	185	oveq2d 7416	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑎 + (1 / 𝑗)) = (𝑎 + (1 / 𝑘)))
187	186	breq2d 5117	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))))
188	187	rabbidv 3424	. . . . . . . 8 ⊢ (𝑗 = 𝑘 → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))})
189	188	eqeq1d 2767	. . . . . . 7 ⊢ (𝑗 = 𝑘 → ({𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚)) ↔ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))))
190	189	rabbidv 3424	. . . . . 6 ⊢ (𝑗 = 𝑘 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
191	184, 190	sylan9eq 2820	. . . . 5 ⊢ ((𝑙 = 𝑚 ∧ 𝑗 = 𝑘) → {𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
192	158, 159, 160, 161, 191	cbvmpo 7494	. . . 4 ⊢ (𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ {𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))}) = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
193	192	eqcomi 2774	. . 3 ⊢ (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}) = (𝑙 ∈ 𝑍, 𝑗 ∈ ℕ ↦ {𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))})
194	128, 16, 129, 130, 93, 148, 149, 193	smflimlem6 47348	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐺‘𝑦) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))
195	1, 2, 42, 126, 194	issmfled 47329	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Ⅎwnfc 2912  ∃wrex 3089  {crab 3417   ∩ cin 3906   ⊆ wss 3907  ∪ cuni 4868  ∪ ciun 4952  ∩ ciin 4953   class class class wbr 5105   ↦ cmpt 5186  dom cdm 5652  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  ℝcr 11087  1c1 11089   + caddc 11091   < clt 11231   / cdiv 11859  ℕcn 12224  ℤcz 12582  ℤ_≥cuz 12853   ⇝ cli 15525  SAlgcsalg 46880  SMblFncsmblfn 47267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-cc 10407  ax-ac2 10435  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-oadd 8445  df-omul 8446  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9913  df-acn 9916  df-ac 10088  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-z 12583  df-uz 12854  df-q 12964  df-rp 13008  df-ioo 13367  df-ico 13369  df-fl 13816  df-seq 14029  df-exp 14089  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-clim 15529  df-rlim 15530  df-rest 17465  df-salg 46881  df-smblfn 47268

This theorem is referenced by:  smflim2  47378