smflimlem2 - Mathbox for Glauco Siliprandi

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Theorem smflimlem2 46162

Description: Lemma for the proof that the limit of sigma-measurable functions is sigma-measurable, Proposition 121F (a) of [Fremlin1] p. 38 . This lemma proves one-side of the double inclusion for the proof that the preimages of right-closed, unbounded-below intervals are in the subspace sigma-algebra induced by 𝐷. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
smflimlem2.1	⊢ 𝑍 = (ℤ_≥‘𝑀)
smflimlem2.2	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimlem2.3	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimlem2.4	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
smflimlem2.5	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
smflimlem2.6	⊢ (𝜑 → 𝐴 ∈ ℝ)
smflimlem2.7	⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
smflimlem2.8	⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
smflimlem2.9	⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)
smflimlem2.10	⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟)

**Assertion**
Ref	Expression
smflimlem2	⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ (𝐷 ∩ 𝐼))

Distinct variable groups: 𝐴,𝑘,𝑚,𝑛 𝐴,𝑠,𝑘,𝑚 𝐶,𝑟 𝐷,𝑘,𝑚,𝑛 𝑛,𝐹,𝑥 𝐹,𝑠,𝑥 𝑘,𝐺,𝑚,𝑛 𝐻,𝑠 𝑥,𝐼 𝑃,𝑟 𝑆,𝑠 𝑘,𝑍,𝑚,𝑛,𝑥 𝜑,𝑘,𝑚,𝑛,𝑥 𝑘,𝑟,𝑚,𝜑 Allowed substitution hints: 𝜑(𝑠) 𝐴(𝑥,𝑟) 𝐶(𝑥,𝑘,𝑚,𝑛,𝑠) 𝐷(𝑥,𝑠,𝑟) 𝑃(𝑥,𝑘,𝑚,𝑛,𝑠) 𝑆(𝑥,𝑘,𝑚,𝑛,𝑟) 𝐹(𝑘,𝑚,𝑟) 𝐺(𝑥,𝑠,𝑟) 𝐻(𝑥,𝑘,𝑚,𝑛,𝑟) 𝐼(𝑘,𝑚,𝑛,𝑠,𝑟) 𝑀(𝑥,𝑘,𝑚,𝑛,𝑠,𝑟) 𝑍(𝑠,𝑟)

Proof of Theorem smflimlem2 Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimlem2.4	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
2		nfrab1 3448	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
3	1, 2	nfcxfr 2896	. . . 4 ⊢ Ⅎ𝑥𝐷
4	3	ssrab2f 44486	. . 3 ⊢ {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ 𝐷
5	4	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ 𝐷)
6		simpllr 774	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ 𝐷)
7		ssrab2 4075	. . . . . . . . . . . . . . 15 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
8	1, 7	eqsstri 4014	. . . . . . . . . . . . . 14 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
9	8	sseli 3976	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
10		fveq2 6900	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑖))
11	10	iineq1d 44459	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑖 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
12	11	cbviunv 5045	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)
13	12	eleq2i 2820	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
14		eliun 5002	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚) ↔ ∃𝑖 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
15	13, 14	bitri 274	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑖 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
16	9, 15	sylib 217	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐷 → ∃𝑖 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
17	6, 16	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) → ∃𝑖 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
18		nfv 1909	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴)
19		nfv 1909	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚 𝑘 ∈ ℕ
20	18, 19	nfan 1894	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ)
21		nfv 1909	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑖 ∈ 𝑍
22	20, 21	nfan 1894	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍)
23		nfcv 2898	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚𝑥
24		nfii1 5034	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)
25	23, 24	nfel 2913	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)
26	22, 25	nfan 1894	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
27		nfmpt1 5258	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
28		eqid 2727	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑖) = (ℤ_≥‘𝑖)
29		uzssz 12879	. . . . . . . . . . . . . . . . . 18 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
30		smflimlem2.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑍 = (ℤ_≥‘𝑀)
31	30	eleq2i 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
32	31	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
33	29, 32	sselid 3978	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℤ)
34		uzid 12873	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ (ℤ_≥‘𝑖))
35	33, 34	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑖))
36	35	ad2antlr 725	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → 𝑖 ∈ (ℤ_≥‘𝑖))
37		simplll 773	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → (𝜑 ∧ 𝑥 ∈ 𝐷))
38	37	simpld 493	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝜑)
39		uzss 12881	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
40	32, 39	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
41	40, 30	sseqtrrdi 4031	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ 𝑍)
42	41	sselda 3980	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑚 ∈ 𝑍)
43	42	ad4ant24 752	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑚 ∈ 𝑍)
44		eliinid 44480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ dom (𝐹‘𝑚))
45	44	adantll 712	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ dom (𝐹‘𝑚))
46		eqidd 2728	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
47		fvexd 6915	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑥) ∈ V)
48	46, 47	fvmpt2d 7021	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
49	48	3adant3 1129	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
50		smflimlem2.2	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝑆 ∈ SAlg)
51	50	adantr 479	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
52		smflimlem2.3	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
53	52	ffvelcdmda 7097	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
54		eqid 2727	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
55	51, 53, 54	smff 46122	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
56	55	3adant3 1129	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
57		simp3 1135	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → 𝑥 ∈ dom (𝐹‘𝑚))
58	56, 57	ffvelcdmd 7098	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
59	49, 58	eqeltrd 2828	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ)
60	38, 43, 45, 59	syl3anc 1368	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ)
61	60	adantl3r 748	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ)
62	61	adantl3r 748	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ)
63	1	eleq2i 2820	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ 𝐷 ↔ 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
64	63	biimpi 215	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
65		rabidim2 44471	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ )
66	64, 65	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 ∈ 𝐷 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ )
67		climdm 15536	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ⇝ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
68	66, 67	sylib 217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 ∈ 𝐷 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ⇝ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
69	68	adantl 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ⇝ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
70	69, 67	sylibr 233	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ )
71	70, 67	sylib 217	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ⇝ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
72		nfcv 2898	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐹
73		smflimlem2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
74		simpr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
75	3, 72, 73, 74	fnlimfv 45053	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
76	75	eqcomd 2733	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (𝐺‘𝑥))
77	71, 76	breqtrd 5176	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ⇝ (𝐺‘𝑥))
78	77	ad4antr 730	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ⇝ (𝐺‘𝑥))
79		smflimlem2.6	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ)
80	79	ad5antr 732	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → 𝐴 ∈ ℝ)
81		simp-4r 782	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → (𝐺‘𝑥) ≤ 𝐴)
82		simpllr 774	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → 𝑘 ∈ ℕ)
83		nnrecrp 44770	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (1 / 𝑘) ∈ ℝ⁺)
84	82, 83	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → (1 / 𝑘) ∈ ℝ⁺)
85	26, 27, 28, 36, 62, 78, 80, 81, 84	climleltrp 45066	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → ∃𝑛 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (ℤ_≥‘𝑛)(((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))))
86		simp-6l 785	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝜑)
87		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → 𝑖 ∈ 𝑍)
88	87	adantr 479	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑖 ∈ 𝑍)
89		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
90		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝑛 ∈ (ℤ_≥‘𝑖))
91		nfv 1909	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚𝜑
92	91, 21, 25	nf3an 1896	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
93		nfv 1909	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚 𝑛 ∈ (ℤ_≥‘𝑖)
94	92, 93	nfan 1894	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖))
95		simpll 765	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)))
96	28	uztrn2 12877	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑖))
97	96	adantll 712	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑖))
98		simpll2 1210	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → 𝑖 ∈ 𝑍)
99		simplr 767	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → 𝑚 ∈ (ℤ_≥‘𝑖))
100	98, 99, 42	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → 𝑚 ∈ 𝑍)
101		simpr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘)))
102		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ 𝑍 → 𝑚 ∈ 𝑍)
103		fvexd 6915	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ 𝑍 → ((𝐹‘𝑚)‘𝑥) ∈ V)
104		eqid 2727	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
105	104	fvmpt2 7019	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ 𝑍 ∧ ((𝐹‘𝑚)‘𝑥) ∈ V) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
106	102, 103, 105	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
107	106	eqcomd 2733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ 𝑍 → ((𝐹‘𝑚)‘𝑥) = ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚))
108	107	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ 𝑍 ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ((𝐹‘𝑚)‘𝑥) = ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚))
109		simpr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ 𝑍 ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘)))
110	108, 109	eqbrtrd 5172	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ 𝑍 ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)))
111	100, 101, 110	syl2anc 582	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)))
112	44	3ad2antl3 1184	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ dom (𝐹‘𝑚))
113	112	adantr 479	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))) → 𝑥 ∈ dom (𝐹‘𝑚))
114		simpr 483	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))) → ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)))
115	113, 114	jca 510	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))) → (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))))
116		rabid 3449	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ↔ (𝑥 ∈ dom (𝐹‘𝑚) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))))
117	115, 116	sylibr 233	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
118	111, 117	syldan 589	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
119	118	adantrl 714	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) ∧ (((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘)))) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
120	119	ex 411	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → ((((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
121	95, 97, 120	syl2anc 582	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
122	94, 121	ralimdaa 3253	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → (∀𝑚 ∈ (ℤ_≥‘𝑛)(((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
123	86, 88, 89, 90, 122	syl31anc 1370	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → (∀𝑚 ∈ (ℤ_≥‘𝑛)(((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
124	123	reximdva 3164	. . . . . . . . . . . . . 14 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → (∃𝑛 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (ℤ_≥‘𝑛)(((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ∃𝑛 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
125	85, 124	mpd 15	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → ∃𝑛 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
126		ssrexv 4049	. . . . . . . . . . . . . . 15 ⊢ ((ℤ_≥‘𝑖) ⊆ 𝑍 → (∃𝑛 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
127	41, 126	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝑍 → (∃𝑛 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
128	127	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → (∃𝑛 ∈ (ℤ_≥‘𝑖)∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
129	125, 128	mpd 15	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
130	129	rexlimdva2 3153	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) → (∃𝑖 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}))
131	17, 130	mpd 15	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
132		nfv 1909	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍)
133		nfra1 3277	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}
134	132, 133	nfan 1894	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
135		simpll1 1209	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
136		simpll2 1210	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ ℕ)
137	30	uztrn2 12877	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
138	137	ssd 44449	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
139	138	sselda 3980	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
140	139	adantll 712	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
141	140	3adantl1 1163	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
142	141	adantlr 713	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
143		rspa 3241	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
144	143	adantll 712	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
145		simp1 1133	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝜑)
146		simp3 1135	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
147		simp2 1134	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑘 ∈ ℕ)
148		eqid 2727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
149	148, 50	rabexd 5337	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
150	149	ralrimivw 3146	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
151	150	ralrimivw 3146	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
152	151	3ad2ant1 1130	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
153		smflimlem2.7	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
154	153	elrnmpoid 44604	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V) → (𝑚𝑃𝑘) ∈ ran 𝑃)
155	146, 147, 152, 154	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) ∈ ran 𝑃)
156		ovex 7457	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚𝑃𝑘) ∈ V
157		eleq1 2816	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝑟 ∈ ran 𝑃 ↔ (𝑚𝑃𝑘) ∈ ran 𝑃))
158	157	anbi2d 628	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝜑 ∧ 𝑟 ∈ ran 𝑃) ↔ (𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃)))
159		fveq2 6900	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝐶‘𝑟) = (𝐶‘(𝑚𝑃𝑘)))
160		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 = (𝑚𝑃𝑘) → 𝑟 = (𝑚𝑃𝑘))
161	159, 160	eleq12d 2822	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝐶‘𝑟) ∈ 𝑟 ↔ (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘)))
162	158, 161	imbi12d 343	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑟 = (𝑚𝑃𝑘) → (((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟) ↔ ((𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))))
163		smflimlem2.10	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑟 ∈ ran 𝑃) → (𝐶‘𝑟) ∈ 𝑟)
164	156, 162, 163	vtocl 3543	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
165	145, 155, 164	syl2anc 582	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
166		fvexd 6915	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ V)
167		smflimlem2.8	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
168	167	ovmpt4g 7572	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ (𝐶‘(𝑚𝑃𝑘)) ∈ V) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
169	146, 147, 166, 168	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
170	169	eqcomd 2733	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) = (𝑚𝐻𝑘))
171	145, 149	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
172	153	ovmpt4g 7572	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V) → (𝑚𝑃𝑘) = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
173	146, 147, 171, 172	syl3anc 1368	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
174	170, 173	eleq12d 2822	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → ((𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘) ↔ (𝑚𝐻𝑘) ∈ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}))
175	165, 174	mpbid 231	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) ∈ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
176		ineq1 4205	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = (𝑚𝐻𝑘) → (𝑠 ∩ dom (𝐹‘𝑚)) = ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚)))
177	176	eqeq2d 2738	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑠 = (𝑚𝐻𝑘) → ({𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)) ↔ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚))))
178	177	elrab 3682	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚𝐻𝑘) ∈ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ↔ ((𝑚𝐻𝑘) ∈ 𝑆 ∧ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚))))
179	175, 178	sylib 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → ((𝑚𝐻𝑘) ∈ 𝑆 ∧ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚))))
180	179	simprd 494	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚)))
181		inss1 4229	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚)) ⊆ (𝑚𝐻𝑘)
182	180, 181	eqsstrdi 4034	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ⊆ (𝑚𝐻𝑘))
183	182	adantr 479	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ⊆ (𝑚𝐻𝑘))
184		simpr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) → 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
185	183, 184	sseldd 3981	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) ∧ 𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) → 𝑥 ∈ (𝑚𝐻𝑘))
186	135, 136, 142, 144, 185	syl31anc 1370	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑥 ∈ (𝑚𝐻𝑘))
187	186	ex 411	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) → (𝑚 ∈ (ℤ_≥‘𝑛) → 𝑥 ∈ (𝑚𝐻𝑘)))
188	134, 187	ralrimi 3250	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) → ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ (𝑚𝐻𝑘))
189		vex 3475	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
190		eliin 5003	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ V → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ (𝑚𝐻𝑘)))
191	189, 190	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ (𝑚𝐻𝑘))
192	188, 191	sylibr 233	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}) → 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘))
193	192	ex 411	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} → 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)))
194	193	ad5ant145 1366	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑛 ∈ 𝑍) → (∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} → 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)))
195	194	reximdva 3164	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) → (∃𝑛 ∈ 𝑍 ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)))
196	131, 195	mpd 15	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) → ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘))
197		eliun 5002	. . . . . . . . 9 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) ↔ ∃𝑛 ∈ 𝑍 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘))
198	196, 197	sylibr 233	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘))
199	198	ralrimiva 3142	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) → ∀𝑘 ∈ ℕ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘))
200		eliin 5003	. . . . . . . 8 ⊢ (𝑥 ∈ V → (𝑥 ∈ ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) ↔ ∀𝑘 ∈ ℕ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)))
201	189, 200	ax-mp 5	. . . . . . 7 ⊢ (𝑥 ∈ ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) ↔ ∀𝑘 ∈ ℕ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘))
202	199, 201	sylibr 233	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) → 𝑥 ∈ ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘))
203		smflimlem2.9	. . . . . 6 ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)
204	202, 203	eleqtrrdi 2839	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) → 𝑥 ∈ 𝐼)
205	204	ex 411	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺‘𝑥) ≤ 𝐴 → 𝑥 ∈ 𝐼))
206	205	ralrimiva 3142	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ((𝐺‘𝑥) ≤ 𝐴 → 𝑥 ∈ 𝐼))
207		rabss 4067	. . 3 ⊢ ({𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ 𝐼 ↔ ∀𝑥 ∈ 𝐷 ((𝐺‘𝑥) ≤ 𝐴 → 𝑥 ∈ 𝐼))
208	206, 207	sylibr 233	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ 𝐼)
209	5, 208	ssind 4233	1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ (𝐷 ∩ 𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3057 ∃wrex 3066 {crab 3428 Vcvv 3471 ∩ cin 3946 ⊆ wss 3947 ∪ ciun 4998 ∩ ciin 4999 class class class wbr 5150 ↦ cmpt 5233 dom cdm 5680 ran crn 5681 ⟶wf 6547 ‘cfv 6551 (class class class)co 7424 ∈ cmpo 7426 ℝcr 11143 1c1 11145 + caddc 11147 < clt 11284 ≤ cle 11285 / cdiv 11907 ℕcn 12248 ℤcz 12594 ℤ_≥cuz 12858 ℝ⁺crp 13012 ⇝ cli 15466 SAlgcsalg 45698 SMblFncsmblfn 46085

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-rep 5287 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-1cn 11202 ax-icn 11203 ax-addcl 11204 ax-addrcl 11205 ax-mulcl 11206 ax-mulrcl 11207 ax-mulcom 11208 ax-addass 11209 ax-mulass 11210 ax-distr 11211 ax-i2m1 11212 ax-1ne0 11213 ax-1rid 11214 ax-rnegex 11215 ax-rrecex 11216 ax-cnre 11217 ax-pre-lttri 11218 ax-pre-lttrn 11219 ax-pre-ltadd 11220 ax-pre-mulgt0 11221 ax-pre-sup 11222

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-iun 5000 df-iin 5001 df-br 5151 df-opab 5213 df-mpt 5234 df-tr 5268 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5635 df-we 5637 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-pred 6308 df-ord 6375 df-on 6376 df-lim 6377 df-suc 6378 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7875 df-1st 7997 df-2nd 7998 df-frecs 8291 df-wrecs 8322 df-recs 8396 df-rdg 8435 df-er 8729 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-sup 9471 df-inf 9472 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-sub 11482 df-neg 11483 df-div 11908 df-nn 12249 df-2 12311 df-3 12312 df-n0 12509 df-z 12595 df-uz 12859 df-rp 13013 df-ioo 13366 df-ico 13368 df-fl 13795 df-seq 14005 df-exp 14065 df-cj 15084 df-re 15085 df-im 15086 df-sqrt 15220 df-abs 15221 df-clim 15470 df-rlim 15471 df-smblfn 46086

This theorem is referenced by: smflimlem5 46165

W3C validator