Theorem smflim 47709

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 1947	. 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 2922	. . . . . . 7 ⊢ Ⅎ𝑥𝑍
5		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑥(ℤ≥‘𝑛)
6		smflim.x	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝐹
7		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑚
8	6, 7	nffv 6883	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐹‘𝑚)
9	8	nfdm 5929	. . . . . . . 8 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
10	5, 9	nfiin 4982	. . . . . . 7 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
11	4, 10	nfiun 4981	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
12	11	ssrab2f 46053	. . . . 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 3976	. . . 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 12956	. . . . . . . . 9 ⊢ (ℤ≥‘𝑀) ⊆ ℤ
16		smflim.z	. . . . . . . . . . 11 ⊢ 𝑍 = (ℤ≥‘𝑀)
17	16	eleq2i 2852	. . . . . . . . . 10 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
18	17	biimpi 219	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
19	15, 18	sselid 3928	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
20		uzid 12950	. . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
21	19, 20	syl 18	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑛))
22	21	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ (ℤ≥‘𝑛))
23	2	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
24		smflim.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
25	24	ffvelcdmda 7072	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹‘𝑛) ∈ (SMblFn‘𝑆))
26		eqid 2760	. . . . . . 7 ⊢ dom (𝐹‘𝑛) = dom (𝐹‘𝑛)
27	23, 25, 26	smfdmss 47665	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐹‘𝑛) ⊆ ∪ 𝑆)
28		smflim.n	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝐹
29		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝑛
30	28, 29	nffv 6883	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝐹‘𝑛)
31	30	nfdm 5929	. . . . . . . 8 ⊢ Ⅎ𝑚dom (𝐹‘𝑛)
32		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑚∪ 𝑆
33	31, 32	nfss 3923	. . . . . . 7 ⊢ Ⅎ𝑚dom (𝐹‘𝑛) ⊆ ∪ 𝑆
34		fveq2 6873	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
35	34	dmeqd 5883	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → dom (𝐹‘𝑚) = dom (𝐹‘𝑛))
36	35	sseq1d 3961	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (dom (𝐹‘𝑚) ⊆ ∪ 𝑆 ↔ dom (𝐹‘𝑛) ⊆ ∪ 𝑆))
37	33, 36	rspce 3565	. . . . . 6 ⊢ ((𝑛 ∈ (ℤ≥‘𝑛) ∧ dom (𝐹‘𝑛) ⊆ ∪ 𝑆) → ∃𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆)
38	22, 27, 37	syl2anc 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∃𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆)
39		iinss 5014	. . . . 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 5008	. . 3 ⊢ (𝜑 → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ ∪ 𝑆)
42	14, 41	sstrd 3940	. 2 ⊢ (𝜑 → 𝐷 ⊆ ∪ 𝑆)
43		nfv 1947	. . . . 5 ⊢ Ⅎ𝑚𝜑
44		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑚𝑦
45		nfmpt1 5203	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
46		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑚dom ⇝
47	45, 46	nfel 2936	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
48		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑍
49		nfii1 4986	. . . . . . . . 9 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
50	48, 49	nfiun 4981	. . . . . . . 8 ⊢ Ⅎ𝑚∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
51	47, 50	nfrabw 3447	. . . . . . 7 ⊢ Ⅎ𝑚{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
52	3, 51	nfcxfr 2920	. . . . . 6 ⊢ Ⅎ𝑚𝐷
53	44, 52	nfel 2936	. . . . 5 ⊢ Ⅎ𝑚 𝑦 ∈ 𝐷
54	43, 53	nfan 1932	. . . 4 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑦 ∈ 𝐷)
55		nfcv 2922	. . . 4 ⊢ Ⅎ𝑤𝐹
56	2	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
57	24	ffvelcdmda 7072	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
58		eqid 2760	. . . . . 6 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
59	56, 57, 58	smff 47664	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
60	59	adantlr 728	. . . 4 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐷) ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
61		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑦∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚)
62		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑦(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
63		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝑦
64	8, 63	nffv 6883	. . . . . . . . 9 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑦)
65	4, 64	nfmpt 5202	. . . . . . . 8 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))
66	65	nfel1 2938	. . . . . . 7 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝
67		fveq2 6873	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
68	67	mpteq2dv 5198	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
69	68	eleq1d 2845	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ))
70	11, 61, 62, 66, 69	cbvrabw 3446	. . . . . 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 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑙dom (𝐹‘𝑚)
72		nfcv 2922	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝑙
73	28, 72	nffv 6883	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(𝐹‘𝑙)
74	73	nfdm 5929	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚dom (𝐹‘𝑙)
75		fveq2 6873	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑙 → (𝐹‘𝑚) = (𝐹‘𝑙))
76	75	dmeqd 5883	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑙 → dom (𝐹‘𝑚) = dom (𝐹‘𝑙))
77	71, 74, 76	cbviin 4993	. . . . . . . . . . . 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 6873	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑖 → (ℤ≥‘𝑛) = (ℤ≥‘𝑖))
80		eqidd 2761	. . . . . . . . . . . 12 ⊢ ((𝑛 = 𝑖 ∧ 𝑙 ∈ (ℤ≥‘𝑖)) → dom (𝐹‘𝑙) = dom (𝐹‘𝑙))
81	79, 80	iineq12dv 46042	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑖 → ∩ 𝑙 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑙) = ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙))
82	78, 81	eqtrd 2795	. . . . . . . . . 10 ⊢ (𝑛 = 𝑖 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙))
83	82	cbviunv 4996	. . . . . . . . 9 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙)
84	83	eleq2i 2852	. . . . . . . 8 ⊢ (𝑦 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑦 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙))
85		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑙𝑍
86		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑙((𝐹‘𝑚)‘𝑦)
87	73, 44	nffv 6883	. . . . . . . . . 10 ⊢ Ⅎ𝑚((𝐹‘𝑙)‘𝑦)
88	75	fveq1d 6875	. . . . . . . . . 10 ⊢ (𝑚 = 𝑙 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑙)‘𝑦))
89	48, 85, 86, 87, 88	cbvmptf 5204	. . . . . . . . 9 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))
90	89	eleq1i 2851	. . . . . . . 8 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)) ∈ dom ⇝ ↔ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ )
91	84, 90	anbi12i 640	. . . . . . 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 3415	. . . . . 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 2787	. . . . 5 ⊢ 𝐷 = {𝑦 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ∣ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ }
94		fveq2 6873	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑙)‘𝑦) = ((𝐹‘𝑙)‘𝑤))
95	94	mpteq2dv 5198	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)))
96	95	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)) ∈ dom ⇝ ↔ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) ∈ dom ⇝ ))
97	96	cbvrabv 3422	. . . . . 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 6873	. . . . . . . . . . . . 13 ⊢ (𝑙 = 𝑚 → (𝐹‘𝑙) = (𝐹‘𝑚))
99	98	dmeqd 5883	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑚 → dom (𝐹‘𝑙) = dom (𝐹‘𝑚))
100	74, 71, 99	cbviin 4993	. . . . . . . . . . 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 4972	. . . . . . . . 9 ⊢ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) = ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚)
103	102	eleq2i 2852	. . . . . . . 8 ⊢ (𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑙 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑙) ↔ 𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚))
104		nfcv 2922	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝑤
105	73, 104	nffv 6883	. . . . . . . . . 10 ⊢ Ⅎ𝑚((𝐹‘𝑙)‘𝑤)
106		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑙((𝐹‘𝑚)‘𝑤)
107	98	fveq1d 6875	. . . . . . . . . 10 ⊢ (𝑙 = 𝑚 → ((𝐹‘𝑙)‘𝑤) = ((𝐹‘𝑚)‘𝑤))
108	85, 48, 105, 106, 107	cbvmptf 5204	. . . . . . . . 9 ⊢ (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤))
109	108	eleq1i 2851	. . . . . . . 8 ⊢ ((𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑤)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)) ∈ dom ⇝ )
110	103, 109	anbi12i 640	. . . . . . 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 3415	. . . . . 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 2783	. . . . 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 2783	. . . 4 ⊢ 𝐷 = {𝑤 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑤)) ∈ dom ⇝ }
114		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → 𝑦 ∈ 𝐷)
115	54, 28, 55, 16, 60, 113, 114	fnlimfvre 46606	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))) ∈ ℝ)
116		smflim.g	. . . 4 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
117		nfrab1 3431	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
118	3, 117	nfcxfr 2920	. . . . 5 ⊢ Ⅎ𝑥𝐷
119		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑦𝐷
120		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑦( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
121		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥 ⇝
122	121, 65	nffv 6883	. . . . 5 ⊢ Ⅎ𝑥( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦)))
123	68	fveq2d 6877	. . . . 5 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
124	118, 119, 120, 122, 123	cbvmptf 5204	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
125	116, 124	eqtri 2783	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑦))))
126	115, 125	fmptd 7102	. 2 ⊢ (𝜑 → 𝐺:𝐷⟶ℝ)
127		smflim.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
128	127	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑀 ∈ ℤ)
129	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg)
130	24	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝐹:𝑍⟶(SMblFn‘𝑆))
131		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑙
132	6, 131	nffv 6883	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐹‘𝑙)
133	132, 63	nffv 6883	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑙)‘𝑦)
134	4, 133	nfmpt 5202	. . . . . 6 ⊢ Ⅎ𝑥(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))
135	121, 134	nffv 6883	. . . . 5 ⊢ Ⅎ𝑥( ⇝ ‘(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)))
136		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑙((𝐹‘𝑚)‘𝑥)
137		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑚𝑥
138	73, 137	nffv 6883	. . . . . . . . 9 ⊢ Ⅎ𝑚((𝐹‘𝑙)‘𝑥)
139	75	fveq1d 6875	. . . . . . . . 9 ⊢ (𝑚 = 𝑙 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑙)‘𝑥))
140	48, 85, 136, 138, 139	cbvmptf 5204	. . . . . . . 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 488	. . . . . . . . 9 ⊢ ((𝑥 = 𝑦 ∧ 𝑙 ∈ 𝑍) → 𝑥 = 𝑦)
143	142	fveq2d 6877	. . . . . . . 8 ⊢ ((𝑥 = 𝑦 ∧ 𝑙 ∈ 𝑍) → ((𝐹‘𝑙)‘𝑥) = ((𝐹‘𝑙)‘𝑦))
144	143	mpteq2dva 5197	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑥)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)))
145	141, 144	eqtrd 2795	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦)))
146	145	fveq2d 6877	. . . . 5 ⊢ (𝑥 = 𝑦 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = ( ⇝ ‘(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))))
147	118, 119, 120, 135, 146	cbvmptf 5204	. . . 4 ⊢ (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))))
148	116, 147	eqtri 2783	. . 3 ⊢ 𝐺 = (𝑦 ∈ 𝐷 ↦ ( ⇝ ‘(𝑙 ∈ 𝑍 ↦ ((𝐹‘𝑙)‘𝑦))))
149		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ)
150		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑚 <
151		nfcv 2922	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑎 + (1 / 𝑗))
152	87, 150, 151	nfbr 5151	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
153	152, 74	nfrabw 3447	. . . . . . 7 ⊢ Ⅎ𝑚{𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))}
154		nfcv 2922	. . . . . . . 8 ⊢ Ⅎ𝑚𝑡
155	154, 74	nfin 4169	. . . . . . 7 ⊢ Ⅎ𝑚(𝑡 ∩ dom (𝐹‘𝑙))
156	153, 155	nfeq 2935	. . . . . 6 ⊢ Ⅎ𝑚{𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))
157		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑚𝑆
158	156, 157	nfrabw 3447	. . . . 5 ⊢ Ⅎ𝑚{𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))}
159		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑘{𝑡 ∈ 𝑆 ∣ {𝑦 ∈ dom (𝐹‘𝑙) ∣ ((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))} = (𝑡 ∩ dom (𝐹‘𝑙))}
160		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑙{𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
161		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑗{𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
162		nfcv 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦dom (𝐹‘𝑙)
163	132	nfdm 5929	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥dom (𝐹‘𝑙)
164		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥 <
165		nfcv 2922	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑎 + (1 / 𝑗))
166	133, 164, 165	nfbr 5151	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗))
167		nfv 1947	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))
168		fveq2 6873	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑙)‘𝑦) = ((𝐹‘𝑙)‘𝑥))
169	168	breq1d 5112	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((𝐹‘𝑙)‘𝑦) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗))))
170	162, 163, 166, 167, 169	cbvrabw 3446	. . . . . . . . . . 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 4158	. . . . . . . . . 10 ⊢ (𝑡 = 𝑠 → (𝑡 ∩ dom (𝐹‘𝑙)) = (𝑠 ∩ dom (𝐹‘𝑙)))
173	171, 172	eqeq12d 2776	. . . . . . . . 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 3422	. . . . . . . 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 2846	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑚 → (𝑥 ∈ dom (𝐹‘𝑙) ↔ 𝑥 ∈ dom (𝐹‘𝑚)))
177	98	fveq1d 6875	. . . . . . . . . . . 12 ⊢ (𝑙 = 𝑚 → ((𝐹‘𝑙)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
178	177	breq1d 5112	. . . . . . . . . . 11 ⊢ (𝑙 = 𝑚 → (((𝐹‘𝑙)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗))))
179	176, 178	anbi12d 644	. . . . . . . . . 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 3414	. . . . . . . . 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 4165	. . . . . . . . 9 ⊢ (𝑙 = 𝑚 → (𝑠 ∩ dom (𝐹‘𝑙)) = (𝑠 ∩ dom (𝐹‘𝑚)))
182	180, 181	eqeq12d 2776	. . . . . . . 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 3419	. . . . . . 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 2795	. . . . . 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 7416	. . . . . . . . . . 11 ⊢ (𝑗 = 𝑘 → (1 / 𝑗) = (1 / 𝑘))
186	185	oveq2d 7424	. . . . . . . . . 10 ⊢ (𝑗 = 𝑘 → (𝑎 + (1 / 𝑗)) = (𝑎 + (1 / 𝑘)))
187	186	breq2d 5114	. . . . . . . . 9 ⊢ (𝑗 = 𝑘 → (((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑗)) ↔ ((𝐹‘𝑚)‘𝑥) < (𝑎 + (1 / 𝑘))))
188	187	rabbidv 3419	. . . . . . . 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 2762	. . . . . . 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 3419	. . . . . 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 2815	. . . . 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 7502	. . . 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 2769	. . 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 47708	. 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑦 ∈ 𝐷 ∣ (𝐺‘𝑦) ≤ 𝑎} ∈ (𝑆 ↾t 𝐷))
195	1, 2, 42, 126, 194	issmfled 47689	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  Ⅎwnfc 2907  ∃wrex 3086  {crab 3412   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4866  ∪ ciun 4950  ∩ ciin 4951   class class class wbr 5102   ↦ cmpt 5185  dom cdm 5647  ⟶wf 6523  ‘cfv 6527  (class class class)co 7408   ∈ cmpo 7410  ℝcr 11171  1c1 11173   + caddc 11175   < clt 11315   / cdiv 11943  ℕcn 12305  ℤcz 12663  ℤ_≥cuz 12935   ⇝ cli 15619  SAlgcsalg 47240  SMblFncsmblfn 47627

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-inf2 9620  ax-cc 10485  ax-cnex 11228  ax-resscn 11229  ax-1cn 11230  ax-icn 11231  ax-addcl 11232  ax-addrcl 11233  ax-mulcl 11234  ax-mulrcl 11235  ax-mulcom 11236  ax-addass 11237  ax-mulass 11238  ax-distr 11239  ax-i2m1 11240  ax-1ne0 11241  ax-1rid 11242  ax-rnegex 11243  ax-rrecex 11244  ax-cnre 11245  ax-pre-lttri 11246  ax-pre-lttrn 11247  ax-pre-ltadd 11248  ax-pre-mulgt0 11249  ax-pre-sup 11250

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-isom 6536  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-1o 8454  df-oadd 8458  df-omul 8459  df-er 8695  df-map 8827  df-pm 8828  df-en 8952  df-dom 8953  df-sdom 8954  df-fin 8955  df-sup 9412  df-inf 9413  df-oi 9482  df-card 9992  df-acn 9995  df-pnf 11317  df-mnf 11318  df-xr 11319  df-ltxr 11320  df-le 11321  df-sub 11515  df-neg 11516  df-div 11944  df-nn 12306  df-2 12375  df-3 12376  df-n0 12577  df-z 12664  df-uz 12936  df-q 13046  df-rp 13091  df-ioo 13450  df-ico 13452  df-fl 13901  df-seq 14114  df-exp 14174  df-cj 15234  df-re 15235  df-im 15236  df-sqrt 15370  df-abs 15371  df-clim 15623  df-rlim 15624  df-rest 17555  df-salg 47241  df-smblfn 47628

This theorem is used by:  smflim2  47738