smflimlem2 - Mathbox for Glauco Siliprandi

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Theorem smflimlem2 44307

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 3317	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
3	1, 2	nfcxfr 2905	. . . 4 ⊢ Ⅎ𝑥𝐷
4	3	ssrab2f 42666	. . 3 ⊢ {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ 𝐷
5	4	a1i 11	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ 𝐷)
6		simpllr 773	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) → 𝑥 ∈ 𝐷)
7		ssrab2 4013	. . . . . . . . . . . . . . 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 3955	. . . . . . . . . . . . . 14 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
9	8	sseli 3917	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝐷 → 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
10		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑖 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑖))
11	10	iineq1d 42640	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑖 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
12	11	cbviunv 4970	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)
13	12	eleq2i 2830	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑖 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
14		eliun 4928	. . . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴)
19		nfv 1917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚 𝑘 ∈ ℕ
20	18, 19	nfan 1902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚(((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ)
21		nfv 1917	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑖 ∈ 𝑍
22	20, 21	nfan 1902	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍)
23		nfcv 2907	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚𝑥
24		nfii1 4959	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)
25	23, 24	nfel 2921	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)
26	22, 25	nfan 1902	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
27		nfmpt1 5182	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
28		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (ℤ_≥‘𝑖) = (ℤ_≥‘𝑖)
29		uzssz 12603	. . . . . . . . . . . . . . . . . 18 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
30		smflimlem2.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑍 = (ℤ_≥‘𝑀)
31	30	eleq2i 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑖 ∈ 𝑍 ↔ 𝑖 ∈ (ℤ_≥‘𝑀))
32	31	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑀))
33	29, 32	sselid 3919	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ ℤ)
34		uzid 12597	. . . . . . . . . . . . . . . . 17 ⊢ (𝑖 ∈ ℤ → 𝑖 ∈ (ℤ_≥‘𝑖))
35	33, 34	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝑖 ∈ 𝑍 → 𝑖 ∈ (ℤ_≥‘𝑖))
36	35	ad2antlr 724	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → 𝑖 ∈ (ℤ_≥‘𝑖))
37		simplll 772	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → (𝜑 ∧ 𝑥 ∈ 𝐷))
38	37	simpld 495	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝜑)
39		uzss 12605	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑖 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
40	32, 39	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ (ℤ_≥‘𝑀))
41	40, 30	sseqtrrdi 3972	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 ∈ 𝑍 → (ℤ_≥‘𝑖) ⊆ 𝑍)
42	41	sselda 3921	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑚 ∈ 𝑍)
43	42	ad4ant24 751	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑚 ∈ 𝑍)
44		eliinid 42661	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ dom (𝐹‘𝑚))
45	44	adantll 711	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ dom (𝐹‘𝑚))
46		eqidd 2739	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))
47		fvexd 6789	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑥) ∈ V)
48	46, 47	fvmpt2d 6888	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
49	48	3adant3 1131	. . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
52		smflimlem2.3	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
53	52	ffvelrnda 6961	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
54		eqid 2738	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
55	51, 53, 54	smff 44268	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
56	55	3adant3 1131	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
57		simp3 1137	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → 𝑥 ∈ dom (𝐹‘𝑚))
58	56, 57	ffvelrnd 6962	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
59	49, 58	eqeltrd 2839	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐹‘𝑚)) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) ∈ ℝ)
60	38, 43, 45, 59	syl3anc 1370	. . . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . . 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 747	. . . . . . . . . . . . . . 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 2830	. . . . . . . . . . . . . . . . . . . . . . 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 42652	. . . . . . . . . . . . . . . . . . . . . 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 15263	. . . . . . . . . . . . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . 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 2907	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑥𝐹
73		smflimlem2.5	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
74		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → 𝑥 ∈ 𝐷)
75	3, 72, 73, 74	fnlimfv 43204	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝐺‘𝑥) = ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
76	75	eqcomd 2744	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (𝐺‘𝑥))
77	71, 76	breqtrd 5100	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ⇝ (𝐺‘𝑥))
78	77	ad4antr 729	. . . . . . . . . . . . . . 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 731	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → 𝐴 ∈ ℝ)
81		simp-4r 781	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → (𝐺‘𝑥) ≤ 𝐴)
82		simpllr 773	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → 𝑘 ∈ ℕ)
83		nnrecrp 42925	. . . . . . . . . . . . . . . 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 43217	. . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) → 𝜑)
87		simplr 766	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) ∧ 𝑘 ∈ ℕ) ∧ 𝑖 ∈ 𝑍) ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) → 𝑖 ∈ 𝑍)
88	87	adantr 481	. . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . 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 1917	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑚𝜑
92	91, 21, 25	nf3an 1904	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚))
93		nfv 1917	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑚 𝑛 ∈ (ℤ_≥‘𝑖)
94	92, 93	nfan 1902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖))
95		simpll 764	. . . . . . . . . . . . . . . . . 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 12601	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑖) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑖))
97	96	adantll 711	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑛 ∈ (ℤ_≥‘𝑖)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑖))
98		simpll2 1212	. . . . . . . . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . 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 6789	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑚 ∈ 𝑍 → ((𝐹‘𝑚)‘𝑥) ∈ V)
104		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
105	104	fvmpt2 6886	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ 𝑍 ∧ ((𝐹‘𝑚)‘𝑥) ∈ V) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
106	102, 103, 105	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚 ∈ 𝑍 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) = ((𝐹‘𝑚)‘𝑥))
107	106	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑚 ∈ 𝑍 → ((𝐹‘𝑚)‘𝑥) = ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚))
108	107	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑚 ∈ 𝑍 ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ((𝐹‘𝑚)‘𝑥) = ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚))
109		simpr 485	. . . . . . . . . . . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 ∈ 𝑍 ∧ ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))‘𝑚) < (𝐴 + (1 / 𝑘))) → ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)))
111	100, 101, 110	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 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 1186	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑖)) → 𝑥 ∈ dom (𝐹‘𝑚))
113	112	adantr 481	. . . . . . . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . . . . . . 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 512	. . . . . . . . . . . . . . . . . . . . . 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 3310	. . . . . . . . . . . . . . . . . . . . . 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 591	. . . . . . . . . . . . . . . . . . . 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 713	. . . . . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . 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 3142	. . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . 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 3203	. . . . . . . . . . . . . 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 3988	. . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 3216	. . . . . . . . . . 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 1917	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍)
133		nfra1 3144	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))}
134	132, 133	nfan 1902	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ ∀𝑚 ∈ (ℤ_≥‘𝑛)𝑥 ∈ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))})
135		simpll1 1211	. . . . . . . . . . . . . . . . 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 1212	. . . . . . . . . . . . . . . . 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 12601	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
138	137	ssd 42630	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
139	138	sselda 3921	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
140	139	adantll 711	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
141	140	3adantl1 1165	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ 𝑍)
142	141	adantlr 712	. . . . . . . . . . . . . . . . 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 3132	. . . . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝜑)
146		simp3 1137	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
147		simp2 1136	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → 𝑘 ∈ ℕ)
148		eqid 2738	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5257	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
150	149	ralrimivw 3104	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
151	150	ralrimivw 3104	. . . . . . . . . . . . . . . . . . . . . . . . . 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 1132	. . . . . . . . . . . . . . . . . . . . . . . . 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 42767	. . . . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝑃𝑘) ∈ ran 𝑃)
156		ovex 7308	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑚𝑃𝑘) ∈ V
157		eleq1 2826	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝑟 ∈ ran 𝑃 ↔ (𝑚𝑃𝑘) ∈ ran 𝑃))
158	157	anbi2d 629	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝜑 ∧ 𝑟 ∈ ran 𝑃) ↔ (𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃)))
159		fveq2 6774	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 = (𝑚𝑃𝑘) → (𝐶‘𝑟) = (𝐶‘(𝑚𝑃𝑘)))
160		id 22	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑟 = (𝑚𝑃𝑘) → 𝑟 = (𝑚𝑃𝑘))
161	159, 160	eleq12d 2833	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑟 = (𝑚𝑃𝑘) → ((𝐶‘𝑟) ∈ 𝑟 ↔ (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘)))
162	158, 161	imbi12d 345	. . . . . . . . . . . . . . . . . . . . . . . . 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 3498	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑚𝑃𝑘) ∈ ran 𝑃) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
165	145, 155, 164	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ (𝑚𝑃𝑘))
166		fvexd 6789	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝐶‘(𝑚𝑃𝑘)) ∈ V)
167		smflimlem2.8	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
168	167	ovmpt4g 7420	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑘 ∈ ℕ ∧ (𝐶‘(𝑚𝑃𝑘)) ∈ V) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
169	146, 147, 166, 168	syl3anc 1370	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → (𝑚𝐻𝑘) = (𝐶‘(𝑚𝑃𝑘)))
170	169	eqcomd 2744	. . . . . . . . . . . . . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . . . . . . . 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 4139	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑠 = (𝑚𝐻𝑘) → (𝑠 ∩ dom (𝐹‘𝑚)) = ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚)))
177	176	eqeq2d 2749	. . . . . . . . . . . . . . . . . . . . . . 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 3624	. . . . . . . . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚)))
181		inss1 4162	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚𝐻𝑘) ∩ dom (𝐹‘𝑚)) ⊆ (𝑚𝐻𝑘)
182	180, 181	eqsstrdi 3975	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ ∧ 𝑚 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} ⊆ (𝑚𝐻𝑘))
183	182	adantr 481	. . . . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . 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 413	. . . . . . . . . . . . . . 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 3141	. . . . . . . . . . . . . 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 3436	. . . . . . . . . . . . . . 15 ⊢ 𝑥 ∈ V
190		eliin 4929	. . . . . . . . . . . . . . 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 413	. . . . . . . . . . . 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 1368	. . . . . . . . . . 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 3203	. . . . . . . . . 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 4928	. . . . . . . . 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 3103	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) → ∀𝑘 ∈ ℕ 𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘))
200		eliin 4929	. . . . . . . 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 2850	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐷) ∧ (𝐺‘𝑥) ≤ 𝐴) → 𝑥 ∈ 𝐼)
205	204	ex 413	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐷) → ((𝐺‘𝑥) ≤ 𝐴 → 𝑥 ∈ 𝐼))
206	205	ralrimiva 3103	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐷 ((𝐺‘𝑥) ≤ 𝐴 → 𝑥 ∈ 𝐼))
207		rabss 4005	. . 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 4166	1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐺‘𝑥) ≤ 𝐴} ⊆ (𝐷 ∩ 𝐼))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 {crab 3068 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 ∪ ciun 4924 ∩ ciin 4925 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 ℝcr 10870 1c1 10872 + caddc 10874 < clt 11009 ≤ cle 11010 / cdiv 11632 ℕcn 11973 ℤcz 12319 ℤ_≥cuz 12582 ℝ⁺crp 12730 ⇝ cli 15193 SAlgcsalg 43849 SMblFncsmblfn 44233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-sup 9201 df-inf 9202 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-ioo 13083 df-ico 13085 df-fl 13512 df-seq 13722 df-exp 13783 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-smblfn 44234

This theorem is referenced by: smflimlem5 44310

W3C validator