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Theorem limciun 23895
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 23876 . . . 4 (𝐹 lim 𝐶) ⊆ ℂ
2 limcresi 23886 . . . . . 6 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
32rgenw 3123 . . . . 5 𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
4 ssiin 4773 . . . . 5 ((𝐹 lim 𝐶) ⊆ 𝑥𝐴 ((𝐹𝐵) lim 𝐶) ↔ ∀𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶))
53, 4mpbir 222 . . . 4 (𝐹 lim 𝐶) ⊆ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)
61, 5ssini 4043 . . 3 (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶))
76a1i 11 . 2 (𝜑 → (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
8 elriin 4796 . . . 4 (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ↔ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)))
9 simprl 778 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ ℂ)
10 limciun.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
1110ad2antrr 708 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → 𝐴 ∈ Fin)
12 simplrr 787 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))
13 nfcv 2959 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
14 nfcsb1v 3755 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎 / 𝑥𝐵
1513, 14nfres 5613 . . . . . . . . . . . . . . . . . . 19 𝑥(𝐹𝑎 / 𝑥𝐵)
16 nfcv 2959 . . . . . . . . . . . . . . . . . . 19 𝑥 lim
17 nfcv 2959 . . . . . . . . . . . . . . . . . . 19 𝑥𝐶
1815, 16, 17nfov 6914 . . . . . . . . . . . . . . . . . 18 𝑥((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
1918nfcri 2953 . . . . . . . . . . . . . . . . 17 𝑥 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
20 csbeq1a 3748 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
2120reseq2d 5611 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐹𝐵) = (𝐹𝑎 / 𝑥𝐵))
2221oveq1d 6899 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝐹𝐵) lim 𝐶) = ((𝐹𝑎 / 𝑥𝐵) lim 𝐶))
2322eleq2d 2882 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝑦 ∈ ((𝐹𝐵) lim 𝐶) ↔ 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)))
2419, 23rspc 3507 . . . . . . . . . . . . . . . 16 (𝑎𝐴 → (∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶) → 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)))
2512, 24mpan9 498 . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
28 ssiun2 4766 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝐴𝑎 / 𝑥𝐵 𝑎𝐴 𝑎 / 𝑥𝐵)
29 nfcv 2959 . . . . . . . . . . . . . . . . . . . . 21 𝑎𝐵
3029, 14, 20cbviun 4760 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐴 𝐵 = 𝑎𝐴 𝑎 / 𝑥𝐵
3128, 30syl6sseqr 3860 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐴𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3231adantl 469 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3327, 32fssresd 6296 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → (𝐹𝑎 / 𝑥𝐵):𝑎 / 𝑥𝐵⟶ℂ)
34 simpr 473 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎𝐴)
35 limciun.2 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑥𝐴 𝐵 ⊆ ℂ)
3635ad2antrr 708 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ⊆ ℂ)
37 nfcv 2959 . . . . . . . . . . . . . . . . . . . 20 𝑥
3814, 37nfss 3802 . . . . . . . . . . . . . . . . . . 19 𝑥𝑎 / 𝑥𝐵 ⊆ ℂ
3920sseq1d 3840 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐵 ⊆ ℂ ↔ 𝑎 / 𝑥𝐵 ⊆ ℂ))
4038, 39rspc 3507 . . . . . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐶 ∈ ℂ)
44 eqid 2817 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4533, 41, 43, 44ellimc2 23878 . . . . . . . . . . . . . . . 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 697 . . . . . . . . . . . . . . 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 223 . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 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 786 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑢 ∈ (TopOpen‘ℂfld))
50 simplrr 787 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑦𝑢)
51 rsp 3128 . . . . . . . . . . . . 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 3165 . . . . . . . . . . 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 2005 . . . . . . . . . . . 12 𝑎𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)
55 nfcv 2959 . . . . . . . . . . . . 13 𝑥(TopOpen‘ℂfld)
56 nfv 2005 . . . . . . . . . . . . . 14 𝑥 𝐶𝑘
57 nfcv 2959 . . . . . . . . . . . . . . . . 17 𝑥𝑘
58 nfcv 2959 . . . . . . . . . . . . . . . . . 18 𝑥{𝐶}
5914, 58nfdif 3941 . . . . . . . . . . . . . . . . 17 𝑥(𝑎 / 𝑥𝐵 ∖ {𝐶})
6057, 59nfin 4028 . . . . . . . . . . . . . . . 16 𝑥(𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6115, 60nfima 5698 . . . . . . . . . . . . . . 15 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
62 nfcv 2959 . . . . . . . . . . . . . . 15 𝑥𝑢
6361, 62nfss 3802 . . . . . . . . . . . . . 14 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢
6456, 63nfan 1990 . . . . . . . . . . . . 13 𝑥(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6555, 64nfrex 3205 . . . . . . . . . . . 12 𝑥𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6620difeq1d 3937 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝐵 ∖ {𝐶}) = (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6766ineq2d 4024 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑘 ∩ (𝐵 ∖ {𝐶})) = (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
6821, 67imaeq12d 5691 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))))
6968sseq1d 3840 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7069anbi2d 616 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7170rexbidv 3251 . . . . . . . . . . . 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, 71cbvral 3367 . . . . . . . . . . 11 (∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∀𝑎𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7353, 72sylibr 225 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
74 eleq2 2885 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (𝐶𝑘𝐶 ∈ (𝑔𝑥)))
75 ineq1 4017 . . . . . . . . . . . . . 14 (𝑘 = (𝑔𝑥) → (𝑘 ∩ (𝐵 ∖ {𝐶})) = ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})))
7675imaeq2d 5690 . . . . . . . . . . . . 13 (𝑘 = (𝑔𝑥) → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
7776sseq1d 3840 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7874, 77anbi12d 618 . . . . . . . . . . 11 (𝑘 = (𝑔𝑥) → ((𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7978ac6sfi 8453 . . . . . . . . . 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 575 . . . . . . . . 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 22821 . . . . . . . . . . . 12 (TopOpen‘ℂfld) ∈ Top
8281a1i 11 . . . . . . . . . . 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))) → (TopOpen‘ℂfld) ∈ Top)
83 frn 6272 . . . . . . . . . . . 12 (𝑔:𝐴⟶(TopOpen‘ℂfld) → ran 𝑔 ⊆ (TopOpen‘ℂfld))
8483ad2antrl 710 . . . . . . . . . . 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))
8511adantr 468 . . . . . . . . . . . 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)
86 ffn 6266 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → 𝑔 Fn 𝐴)
8786ad2antrl 710 . . . . . . . . . . . . 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)
88 dffn4 6347 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴𝑔:𝐴onto→ran 𝑔)
8987, 88sylib 209 . . . . . . . . . . . 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)
90 fofi 8501 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑔:𝐴onto→ran 𝑔) → ran 𝑔 ∈ Fin)
9185, 89, 90syl2anc 575 . . . . . . . . . . 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)
9244cnfldtopon 22820 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
9392toponunii 20955 . . . . . . . . . . . 12 ℂ = (TopOpen‘ℂfld)
9493rintopn 20948 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ ran 𝑔 ⊆ (TopOpen‘ℂfld) ∧ ran 𝑔 ∈ Fin) → (ℂ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld))
9582, 84, 91, 94syl3anc 1483 . . . . . . . . . 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))
9642adantr 468 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝐶 ∈ ℂ)
9796ad2antrr 708 . . . . . . . . . . 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)
98 simpl 470 . . . . . . . . . . . . . 14 ((𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → 𝐶 ∈ (𝑔𝑥))
9998ralimi 3151 . . . . . . . . . . . . 13 (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥))
10099ad2antll 711 . . . . . . . . . . . 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))
101 eleq2 2885 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑥) → (𝐶𝑧𝐶 ∈ (𝑔𝑥)))
102101ralrn 6594 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴 → (∀𝑧 ∈ ran 𝑔 𝐶𝑧 ↔ ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥)))
10387, 102syl 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)))
104100, 103mpbird 248 . . . . . . . . . . 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)
105 elrint 4721 . . . . . . . . . . 11 (𝐶 ∈ (ℂ ∩ ran 𝑔) ↔ (𝐶 ∈ ℂ ∧ ∀𝑧 ∈ ran 𝑔 𝐶𝑧))
10697, 104, 105sylanbrc 574 . . . . . . . . . 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))
107 indifcom 4085 . . . . . . . . . . . . . 14 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
108 iunin1 4788 . . . . . . . . . . . . . 14 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
109107, 108eqtr4i 2842 . . . . . . . . . . . . 13 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
110109imaeq2i 5688 . . . . . . . . . . . 12 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
111 imaiun 6737 . . . . . . . . . . . 12 (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
112110, 111eqtri 2839 . . . . . . . . . . 11 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
113 inss2 4041 . . . . . . . . . . . . . . . . . . . . 21 (ℂ ∩ ran 𝑔) ⊆ ran 𝑔
114 fnfvelrn 6588 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 Fn 𝐴𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
11586, 114sylan 571 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
116 intss1 4695 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔𝑥) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑥))
117115, 116syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ran 𝑔 ⊆ (𝑔𝑥))
118113, 117syl5ss 3820 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (ℂ ∩ ran 𝑔) ⊆ (𝑔𝑥))
119118ssdifd 3956 . . . . . . . . . . . . . . . . . . 19 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}))
120 sslin 4046 . . . . . . . . . . . . . . . . . . 19 (((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}) → (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
121 imass2 5725 . . . . . . . . . . . . . . . . . . 19 ((𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))))
122119, 120, 1213syl 18 . . . . . . . . . . . . . . . . . 18 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))))
123 indifcom 4085 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})) = (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))
124123imaeq2i 5688 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
125 inss1 4040 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) ⊆ 𝐵
126 resima2 5649 . . . . . . . . . . . . . . . . . . . 20 ((𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) ⊆ 𝐵 → ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))))
127125, 126ax-mp 5 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
128124, 127eqtri 2839 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
129122, 128syl6sseqr 3860 . . . . . . . . . . . . . . . . 17 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
130 sstr2 3816 . . . . . . . . . . . . . . . . 17 ((𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) → (((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
131129, 130syl 17 . . . . . . . . . . . . . . . 16 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
132131adantld 480 . . . . . . . . . . . . . . 15 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
133132ralimdva 3161 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
134133imp 395 . . . . . . . . . . . . 13 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
135134adantl 469 . . . . . . . . . . . 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)
136 iunss 4764 . . . . . . . . . . . 12 ( 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢 ↔ ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
137135, 136sylibr 225 . . . . . . . . . . 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)
138112, 137syl5eqss 3857 . . . . . . . . . 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)
139 eleq2 2885 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐶𝑣𝐶 ∈ (ℂ ∩ ran 𝑔)))
140 ineq1 4017 . . . . . . . . . . . . . 14 (𝑣 = (ℂ ∩ ran 𝑔) → (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})))
141140imaeq2d 5690 . . . . . . . . . . . . 13 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))))
142141sseq1d 3840 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → ((𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
143139, 142anbi12d 618 . . . . . . . . . . 11 (𝑣 = (ℂ ∩ ran 𝑔) → ((𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ (ℂ ∩ ran 𝑔) ∧ (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
144143rspcev 3513 . . . . . . . . . 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))
14595, 106, 138, 144syl12anc 856 . . . . . . . . 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))
14680, 145exlimddv 2026 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
147146expr 446 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
148147ralrimiva 3165 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
14926adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
150 iunss 4764 . . . . . . . . 9 ( 𝑥𝐴 𝐵 ⊆ ℂ ↔ ∀𝑥𝐴 𝐵 ⊆ ℂ)
15135, 150sylibr 225 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ⊆ ℂ)
152151adantr 468 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑥𝐴 𝐵 ⊆ ℂ)
153149, 152, 96, 44ellimc2 23878 . . . . . 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)))))
1549, 148, 153mpbir2and 695 . . . . 5 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ (𝐹 lim 𝐶))
155154ex 399 . . . 4 (𝜑 → ((𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
1568, 155syl5bi 233 . . 3 (𝜑 → (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
157156ssrdv 3815 . 2 (𝜑 → (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ⊆ (𝐹 lim 𝐶))
1587, 157eqssd 3826 1 (𝜑 → (𝐹 lim 𝐶) = (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wex 1859  wcel 2157  wral 3107  wrex 3108  csb 3739  cdif 3777  cin 3779  wss 3780  {csn 4381   cint 4680   ciun 4723   ciin 4724  ran crn 5325  cres 5326  cima 5327   Fn wfn 6106  wf 6107  ontowfo 6109  cfv 6111  (class class class)co 6884  Fincfn 8202  cc 10229  TopOpenctopn 16307  fldccnfld 19974  Topctop 20932   lim climc 23863
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-rep 4977  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189  ax-cnex 10287  ax-resscn 10288  ax-1cn 10289  ax-icn 10290  ax-addcl 10291  ax-addrcl 10292  ax-mulcl 10293  ax-mulrcl 10294  ax-mulcom 10295  ax-addass 10296  ax-mulass 10297  ax-distr 10298  ax-i2m1 10299  ax-1ne0 10300  ax-1rid 10301  ax-rnegex 10302  ax-rrecex 10303  ax-cnre 10304  ax-pre-lttri 10305  ax-pre-lttrn 10306  ax-pre-ltadd 10307  ax-pre-mulgt0 10308  ax-pre-sup 10309
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-pss 3796  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-tp 4386  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-iin 4726  df-br 4856  df-opab 4918  df-mpt 4935  df-tr 4958  df-id 5232  df-eprel 5237  df-po 5245  df-so 5246  df-fr 5283  df-we 5285  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-rn 5335  df-res 5336  df-ima 5337  df-pred 5907  df-ord 5953  df-on 5954  df-lim 5955  df-suc 5956  df-iota 6074  df-fun 6113  df-fn 6114  df-f 6115  df-f1 6116  df-fo 6117  df-f1o 6118  df-fv 6119  df-riota 6845  df-ov 6887  df-oprab 6888  df-mpt2 6889  df-om 7306  df-1st 7408  df-2nd 7409  df-wrecs 7652  df-recs 7714  df-rdg 7752  df-1o 7806  df-oadd 7810  df-er 7989  df-map 8104  df-pm 8105  df-en 8203  df-dom 8204  df-sdom 8205  df-fin 8206  df-fi 8566  df-sup 8597  df-inf 8598  df-pnf 10371  df-mnf 10372  df-xr 10373  df-ltxr 10374  df-le 10375  df-sub 10563  df-neg 10564  df-div 10980  df-nn 11316  df-2 11376  df-3 11377  df-4 11378  df-5 11379  df-6 11380  df-7 11381  df-8 11382  df-9 11383  df-n0 11580  df-z 11664  df-dec 11780  df-uz 11925  df-q 12028  df-rp 12067  df-xneg 12182  df-xadd 12183  df-xmul 12184  df-fz 12570  df-seq 13045  df-exp 13104  df-cj 14082  df-re 14083  df-im 14084  df-sqrt 14218  df-abs 14219  df-struct 16090  df-ndx 16091  df-slot 16092  df-base 16094  df-plusg 16186  df-mulr 16187  df-starv 16188  df-tset 16192  df-ple 16193  df-ds 16195  df-unif 16196  df-rest 16308  df-topn 16309  df-topgen 16329  df-psmet 19966  df-xmet 19967  df-met 19968  df-bl 19969  df-mopn 19970  df-cnfld 19975  df-top 20933  df-topon 20950  df-topsp 20972  df-bases 20985  df-cnp 21267  df-xms 22359  df-ms 22360  df-limc 23867
This theorem is referenced by:  limcun  23896
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