MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  limciun Structured version   Visualization version   GIF version

Theorem limciun 25056
Description: A point is a limit of 𝐹 on the finite union 𝑥𝐴𝐵(𝑥) iff it is the limit of the restriction of 𝐹 to each 𝐵(𝑥). (Contributed by Mario Carneiro, 30-Dec-2016.)
Hypotheses
Ref Expression
limciun.1 (𝜑𝐴 ∈ Fin)
limciun.2 (𝜑 → ∀𝑥𝐴 𝐵 ⊆ ℂ)
limciun.3 (𝜑𝐹: 𝑥𝐴 𝐵⟶ℂ)
limciun.4 (𝜑𝐶 ∈ ℂ)
Assertion
Ref Expression
limciun (𝜑 → (𝐹 lim 𝐶) = (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐹
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem limciun
Dummy variables 𝑔 𝑎 𝑘 𝑢 𝑣 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limccl 25037 . . . 4 (𝐹 lim 𝐶) ⊆ ℂ
2 limcresi 25047 . . . . . 6 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
32rgenw 3078 . . . . 5 𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
4 ssiin 4990 . . . . 5 ((𝐹 lim 𝐶) ⊆ 𝑥𝐴 ((𝐹𝐵) lim 𝐶) ↔ ∀𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶))
53, 4mpbir 230 . . . 4 (𝐹 lim 𝐶) ⊆ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)
61, 5ssini 4171 . . 3 (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶))
76a1i 11 . 2 (𝜑 → (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
8 elriin 5015 . . . 4 (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ↔ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)))
9 simprl 768 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ ℂ)
10 limciun.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
1110ad2antrr 723 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → 𝐴 ∈ Fin)
12 simplrr 775 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))
13 nfcv 2909 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
14 nfcsb1v 3862 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎 / 𝑥𝐵
1513, 14nfres 5892 . . . . . . . . . . . . . . . . . . 19 𝑥(𝐹𝑎 / 𝑥𝐵)
16 nfcv 2909 . . . . . . . . . . . . . . . . . . 19 𝑥 lim
17 nfcv 2909 . . . . . . . . . . . . . . . . . . 19 𝑥𝐶
1815, 16, 17nfov 7301 . . . . . . . . . . . . . . . . . 18 𝑥((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
1918nfcri 2896 . . . . . . . . . . . . . . . . 17 𝑥 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
20 csbeq1a 3851 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
2120reseq2d 5890 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐹𝐵) = (𝐹𝑎 / 𝑥𝐵))
2221oveq1d 7286 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝐹𝐵) lim 𝐶) = ((𝐹𝑎 / 𝑥𝐵) lim 𝐶))
2322eleq2d 2826 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝑦 ∈ ((𝐹𝐵) lim 𝐶) ↔ 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)))
2419, 23rspc 3548 . . . . . . . . . . . . . . . 16 (𝑎𝐴 → (∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶) → 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)))
2512, 24mpan9 507 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶))
26 limciun.3 . . . . . . . . . . . . . . . . . . 19 (𝜑𝐹: 𝑥𝐴 𝐵⟶ℂ)
2726ad2antrr 723 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
28 ssiun2 4982 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝐴𝑎 / 𝑥𝐵 𝑎𝐴 𝑎 / 𝑥𝐵)
29 nfcv 2909 . . . . . . . . . . . . . . . . . . . . 21 𝑎𝐵
3029, 14, 20cbviun 4971 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐴 𝐵 = 𝑎𝐴 𝑎 / 𝑥𝐵
3128, 30sseqtrrdi 3977 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐴𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3231adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3327, 32fssresd 6639 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → (𝐹𝑎 / 𝑥𝐵):𝑎 / 𝑥𝐵⟶ℂ)
34 simpr 485 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎𝐴)
35 limciun.2 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑥𝐴 𝐵 ⊆ ℂ)
3635ad2antrr 723 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ⊆ ℂ)
37 nfcv 2909 . . . . . . . . . . . . . . . . . . . 20 𝑥
3814, 37nfss 3918 . . . . . . . . . . . . . . . . . . 19 𝑥𝑎 / 𝑥𝐵 ⊆ ℂ
3920sseq1d 3957 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐵 ⊆ ℂ ↔ 𝑎 / 𝑥𝐵 ⊆ ℂ))
4038, 39rspc 3548 . . . . . . . . . . . . . . . . . 18 (𝑎𝐴 → (∀𝑥𝐴 𝐵 ⊆ ℂ → 𝑎 / 𝑥𝐵 ⊆ ℂ))
4134, 36, 40sylc 65 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎 / 𝑥𝐵 ⊆ ℂ)
42 limciun.4 . . . . . . . . . . . . . . . . . 18 (𝜑𝐶 ∈ ℂ)
4342ad2antrr 723 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐶 ∈ ℂ)
44 eqid 2740 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4533, 41, 43, 44ellimc2 25039 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → (𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
4645adantlr 712 . . . . . . . . . . . . . . 15 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → (𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
4725, 46mpbid 231 . . . . . . . . . . . . . 14 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))))
4847simprd 496 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
49 simplrl 774 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑢 ∈ (TopOpen‘ℂfld))
50 simplrr 775 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑦𝑢)
51 rsp 3132 . . . . . . . . . . . . 13 (∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → (𝑢 ∈ (TopOpen‘ℂfld) → (𝑦𝑢 → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))))
5248, 49, 50, 51syl3c 66 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
5352ralrimiva 3110 . . . . . . . . . . 11 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑎𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
54 nfv 1921 . . . . . . . . . . . 12 𝑎𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)
55 nfcv 2909 . . . . . . . . . . . . 13 𝑥(TopOpen‘ℂfld)
56 nfv 1921 . . . . . . . . . . . . . 14 𝑥 𝐶𝑘
57 nfcv 2909 . . . . . . . . . . . . . . . . 17 𝑥𝑘
58 nfcv 2909 . . . . . . . . . . . . . . . . . 18 𝑥{𝐶}
5914, 58nfdif 4065 . . . . . . . . . . . . . . . . 17 𝑥(𝑎 / 𝑥𝐵 ∖ {𝐶})
6057, 59nfin 4156 . . . . . . . . . . . . . . . 16 𝑥(𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6115, 60nfima 5976 . . . . . . . . . . . . . . 15 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
62 nfcv 2909 . . . . . . . . . . . . . . 15 𝑥𝑢
6361, 62nfss 3918 . . . . . . . . . . . . . 14 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢
6456, 63nfan 1906 . . . . . . . . . . . . 13 𝑥(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6555, 64nfrex 3240 . . . . . . . . . . . 12 𝑥𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6620difeq1d 4061 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝐵 ∖ {𝐶}) = (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6766ineq2d 4152 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑘 ∩ (𝐵 ∖ {𝐶})) = (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
6821, 67imaeq12d 5969 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))))
6968sseq1d 3957 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7069anbi2d 629 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7170rexbidv 3228 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∃𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7254, 65, 71cbvralw 3372 . . . . . . . . . . 11 (∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∀𝑎𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7353, 72sylibr 233 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
74 eleq2 2829 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (𝐶𝑘𝐶 ∈ (𝑔𝑥)))
75 ineq1 4145 . . . . . . . . . . . . . 14 (𝑘 = (𝑔𝑥) → (𝑘 ∩ (𝐵 ∖ {𝐶})) = ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})))
7675imaeq2d 5968 . . . . . . . . . . . . 13 (𝑘 = (𝑔𝑥) → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
7776sseq1d 3957 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7874, 77anbi12d 631 . . . . . . . . . . 11 (𝑘 = (𝑔𝑥) → ((𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7978ac6sfi 9036 . . . . . . . . . 10 ((𝐴 ∈ Fin ∧ ∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∃𝑔(𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
8011, 73, 79syl2anc 584 . . . . . . . . 9 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∃𝑔(𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
8144cnfldtop 23945 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ Top
82 frn 6605 . . . . . . . . . . . 12 (𝑔:𝐴⟶(TopOpen‘ℂfld) → ran 𝑔 ⊆ (TopOpen‘ℂfld))
8382ad2antrl 725 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ran 𝑔 ⊆ (TopOpen‘ℂfld))
8411adantr 481 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐴 ∈ Fin)
85 ffn 6598 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → 𝑔 Fn 𝐴)
8685ad2antrl 725 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝑔 Fn 𝐴)
87 dffn4 6692 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴𝑔:𝐴onto→ran 𝑔)
8886, 87sylib 217 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝑔:𝐴onto→ran 𝑔)
89 fofi 9083 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑔:𝐴onto→ran 𝑔) → ran 𝑔 ∈ Fin)
9084, 88, 89syl2anc 584 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ran 𝑔 ∈ Fin)
91 unicntop 23947 . . . . . . . . . . . 12 ℂ = (TopOpen‘ℂfld)
9291rintopn 22056 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ ran 𝑔 ⊆ (TopOpen‘ℂfld) ∧ ran 𝑔 ∈ Fin) → (ℂ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld))
9381, 83, 90, 92mp3an2i 1465 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → (ℂ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld))
9442adantr 481 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝐶 ∈ ℂ)
9594ad2antrr 723 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐶 ∈ ℂ)
96 simpl 483 . . . . . . . . . . . . . 14 ((𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → 𝐶 ∈ (𝑔𝑥))
9796ralimi 3089 . . . . . . . . . . . . 13 (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥))
9897ad2antll 726 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥))
99 eleq2 2829 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑥) → (𝐶𝑧𝐶 ∈ (𝑔𝑥)))
10099ralrn 6961 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴 → (∀𝑧 ∈ ran 𝑔 𝐶𝑧 ↔ ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥)))
10186, 100syl 17 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → (∀𝑧 ∈ ran 𝑔 𝐶𝑧 ↔ ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥)))
10298, 101mpbird 256 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑧 ∈ ran 𝑔 𝐶𝑧)
103 elrint 4928 . . . . . . . . . . 11 (𝐶 ∈ (ℂ ∩ ran 𝑔) ↔ (𝐶 ∈ ℂ ∧ ∀𝑧 ∈ ran 𝑔 𝐶𝑧))
10495, 102, 103sylanbrc 583 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝐶 ∈ (ℂ ∩ ran 𝑔))
105 indifcom 4212 . . . . . . . . . . . . . 14 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
106 iunin1 5006 . . . . . . . . . . . . . 14 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
107105, 106eqtr4i 2771 . . . . . . . . . . . . 13 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
108107imaeq2i 5966 . . . . . . . . . . . 12 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
109 imaiun 7115 . . . . . . . . . . . 12 (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
110108, 109eqtri 2768 . . . . . . . . . . 11 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
111 inss2 4169 . . . . . . . . . . . . . . . . . . . . 21 (ℂ ∩ ran 𝑔) ⊆ ran 𝑔
112 fnfvelrn 6955 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 Fn 𝐴𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
11385, 112sylan 580 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
114 intss1 4900 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔𝑥) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑥))
115113, 114syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ran 𝑔 ⊆ (𝑔𝑥))
116111, 115sstrid 3937 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (ℂ ∩ ran 𝑔) ⊆ (𝑔𝑥))
117116ssdifd 4080 . . . . . . . . . . . . . . . . . . 19 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}))
118 sslin 4174 . . . . . . . . . . . . . . . . . . 19 (((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}) → (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
119 imass2 6009 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))))
120117, 118, 1193syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))))
121 indifcom 4212 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})) = (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))
122121imaeq2i 5966 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
123 inss1 4168 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) ⊆ 𝐵
124 resima2 5925 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) ⊆ 𝐵 → ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))))
125123, 124ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
126122, 125eqtri 2768 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
127120, 126sseqtrrdi 3977 . . . . . . . . . . . . . . . . 17 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
128 sstr2 3933 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) → (((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
129127, 128syl 17 . . . . . . . . . . . . . . . 16 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
130129adantld 491 . . . . . . . . . . . . . . 15 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
131130ralimdva 3105 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
132131imp 407 . . . . . . . . . . . . 13 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
133132adantl 482 . . . . . . . . . . . 12 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
134 iunss 4980 . . . . . . . . . . . 12 ( 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢 ↔ ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
135133, 134sylibr 233 . . . . . . . . . . 11 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
136110, 135eqsstrid 3974 . . . . . . . . . 10 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)
137 eleq2 2829 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐶𝑣𝐶 ∈ (ℂ ∩ ran 𝑔)))
138 ineq1 4145 . . . . . . . . . . . . . 14 (𝑣 = (ℂ ∩ ran 𝑔) → (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})))
139138imaeq2d 5968 . . . . . . . . . . . . 13 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))))
140139sseq1d 3957 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → ((𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
141137, 140anbi12d 631 . . . . . . . . . . 11 (𝑣 = (ℂ ∩ ran 𝑔) → ((𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ (ℂ ∩ ran 𝑔) ∧ (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
142141rspcev 3561 . . . . . . . . . 10 (((ℂ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld) ∧ (𝐶 ∈ (ℂ ∩ ran 𝑔) ∧ (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
14393, 104, 136, 142syl12anc 834 . . . . . . . . 9 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ (𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
14480, 143exlimddv 1942 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
145144expr 457 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
146145ralrimiva 3110 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
14726adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
148 iunss 4980 . . . . . . . . 9 ( 𝑥𝐴 𝐵 ⊆ ℂ ↔ ∀𝑥𝐴 𝐵 ⊆ ℂ)
14935, 148sylibr 233 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ⊆ ℂ)
150149adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑥𝐴 𝐵 ⊆ ℂ)
151147, 150, 94, 44ellimc2 25039 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → (𝑦 ∈ (𝐹 lim 𝐶) ↔ (𝑦 ∈ ℂ ∧ ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))))
1529, 146, 151mpbir2and 710 . . . . 5 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ (𝐹 lim 𝐶))
153152ex 413 . . . 4 (𝜑 → ((𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
1548, 153syl5bi 241 . . 3 (𝜑 → (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
155154ssrdv 3932 . 2 (𝜑 → (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ⊆ (𝐹 lim 𝐶))
1567, 155eqssd 3943 1 (𝜑 → (𝐹 lim 𝐶) = (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1542  wex 1786  wcel 2110  wral 3066  wrex 3067  csb 3837  cdif 3889  cin 3891  wss 3892  {csn 4567   cint 4885   ciun 4930   ciin 4931  ran crn 5591  cres 5592  cima 5593   Fn wfn 6427  wf 6428  ontowfo 6430  cfv 6432  (class class class)co 7271  Fincfn 8716  cc 10870  TopOpenctopn 17130  fldccnfld 20595  Topctop 22040   lim climc 25024
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949  ax-pre-sup 10950
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-iin 4933  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-er 8481  df-map 8600  df-pm 8601  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-fi 9148  df-sup 9179  df-inf 9180  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-z 12320  df-dec 12437  df-uz 12582  df-q 12688  df-rp 12730  df-xneg 12847  df-xadd 12848  df-xmul 12849  df-fz 13239  df-seq 13720  df-exp 13781  df-cj 14808  df-re 14809  df-im 14810  df-sqrt 14944  df-abs 14945  df-struct 16846  df-slot 16881  df-ndx 16893  df-base 16911  df-plusg 16973  df-mulr 16974  df-starv 16975  df-tset 16979  df-ple 16980  df-ds 16982  df-unif 16983  df-rest 17131  df-topn 17132  df-topgen 17152  df-psmet 20587  df-xmet 20588  df-met 20589  df-bl 20590  df-mopn 20591  df-cnfld 20596  df-top 22041  df-topon 22058  df-topsp 22080  df-bases 22094  df-cnp 22377  df-xms 23471  df-ms 23472  df-limc 25028
This theorem is referenced by:  limcun  25057
  Copyright terms: Public domain W3C validator