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Theorem limciun 24163
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 24144 . . . 4 (𝐹 lim 𝐶) ⊆ ℂ
2 limcresi 24154 . . . . . 6 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
32rgenw 3115 . . . . 5 𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
4 ssiin 4872 . . . . 5 ((𝐹 lim 𝐶) ⊆ 𝑥𝐴 ((𝐹𝐵) lim 𝐶) ↔ ∀𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶))
53, 4mpbir 232 . . . 4 (𝐹 lim 𝐶) ⊆ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)
61, 5ssini 4123 . . 3 (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶))
76a1i 11 . 2 (𝜑 → (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
8 elriin 4896 . . . 4 (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ↔ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)))
9 simprl 767 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ ℂ)
10 limciun.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
1110ad2antrr 722 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → 𝐴 ∈ Fin)
12 simplrr 774 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))
13 nfcv 2947 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
14 nfcsb1v 3828 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎 / 𝑥𝐵
1513, 14nfres 5728 . . . . . . . . . . . . . . . . . . 19 𝑥(𝐹𝑎 / 𝑥𝐵)
16 nfcv 2947 . . . . . . . . . . . . . . . . . . 19 𝑥 lim
17 nfcv 2947 . . . . . . . . . . . . . . . . . . 19 𝑥𝐶
1815, 16, 17nfov 7037 . . . . . . . . . . . . . . . . . 18 𝑥((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
1918nfcri 2941 . . . . . . . . . . . . . . . . 17 𝑥 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
20 csbeq1a 3819 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
2120reseq2d 5726 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐹𝐵) = (𝐹𝑎 / 𝑥𝐵))
2221oveq1d 7022 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝐹𝐵) lim 𝐶) = ((𝐹𝑎 / 𝑥𝐵) lim 𝐶))
2322eleq2d 2866 . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
28 ssiun2 4864 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝐴𝑎 / 𝑥𝐵 𝑎𝐴 𝑎 / 𝑥𝐵)
29 nfcv 2947 . . . . . . . . . . . . . . . . . . . . 21 𝑎𝐵
3029, 14, 20cbviun 4858 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐴 𝐵 = 𝑎𝐴 𝑎 / 𝑥𝐵
3128, 30syl6sseqr 3934 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐴𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3231adantl 482 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3327, 32fssresd 6405 . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ⊆ ℂ)
37 nfcv 2947 . . . . . . . . . . . . . . . . . . . 20 𝑥
3814, 37nfss 3877 . . . . . . . . . . . . . . . . . . 19 𝑥𝑎 / 𝑥𝐵 ⊆ ℂ
3920sseq1d 3914 . . . . . . . . . . . . . . . . . . 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 722 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐶 ∈ ℂ)
44 eqid 2793 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4533, 41, 43, 44ellimc2 24146 . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . 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 233 . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑢 ∈ (TopOpen‘ℂfld))
50 simplrr 774 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑦𝑢)
51 rsp 3170 . . . . . . . . . . . . 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 3147 . . . . . . . . . . 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 1890 . . . . . . . . . . . 12 𝑎𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)
55 nfcv 2947 . . . . . . . . . . . . 13 𝑥(TopOpen‘ℂfld)
56 nfv 1890 . . . . . . . . . . . . . 14 𝑥 𝐶𝑘
57 nfcv 2947 . . . . . . . . . . . . . . . . 17 𝑥𝑘
58 nfcv 2947 . . . . . . . . . . . . . . . . . 18 𝑥{𝐶}
5914, 58nfdif 4018 . . . . . . . . . . . . . . . . 17 𝑥(𝑎 / 𝑥𝐵 ∖ {𝐶})
6057, 59nfin 4108 . . . . . . . . . . . . . . . 16 𝑥(𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6115, 60nfima 5806 . . . . . . . . . . . . . . 15 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
62 nfcv 2947 . . . . . . . . . . . . . . 15 𝑥𝑢
6361, 62nfss 3877 . . . . . . . . . . . . . 14 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢
6456, 63nfan 1879 . . . . . . . . . . . . 13 𝑥(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6555, 64nfrex 3268 . . . . . . . . . . . 12 𝑥𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6620difeq1d 4014 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝐵 ∖ {𝐶}) = (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6766ineq2d 4104 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑘 ∩ (𝐵 ∖ {𝐶})) = (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
6821, 67imaeq12d 5799 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))))
6968sseq1d 3914 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7069anbi2d 628 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7170rexbidv 3257 . . . . . . . . . . . 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 3396 . . . . . . . . . . 11 (∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∀𝑎𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7353, 72sylibr 235 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
74 eleq2 2869 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (𝐶𝑘𝐶 ∈ (𝑔𝑥)))
75 ineq1 4096 . . . . . . . . . . . . . 14 (𝑘 = (𝑔𝑥) → (𝑘 ∩ (𝐵 ∖ {𝐶})) = ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})))
7675imaeq2d 5798 . . . . . . . . . . . . 13 (𝑘 = (𝑔𝑥) → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
7776sseq1d 3914 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7874, 77anbi12d 630 . . . . . . . . . . 11 (𝑘 = (𝑔𝑥) → ((𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7978ac6sfi 8598 . . . . . . . . . 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 23063 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ Top
82 frn 6380 . . . . . . . . . . . 12 (𝑔:𝐴⟶(TopOpen‘ℂfld) → ran 𝑔 ⊆ (TopOpen‘ℂfld))
8382ad2antrl 724 . . . . . . . . . . 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 6374 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → 𝑔 Fn 𝐴)
8685ad2antrl 724 . . . . . . . . . . . . 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 6456 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴𝑔:𝐴onto→ran 𝑔)
8886, 87sylib 219 . . . . . . . . . . . 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 8646 . . . . . . . . . . . 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 23065 . . . . . . . . . . . 12 ℂ = (TopOpen‘ℂfld)
9291rintopn 21189 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ ran 𝑔 ⊆ (TopOpen‘ℂfld) ∧ ran 𝑔 ∈ Fin) → (ℂ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld))
9381, 83, 90, 92mp3an2i 1456 . . . . . . . . . 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 722 . . . . . . . . . . 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 3125 . . . . . . . . . . . . 13 (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥))
9897ad2antll 725 . . . . . . . . . . . 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 2869 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑥) → (𝐶𝑧𝐶 ∈ (𝑔𝑥)))
10099ralrn 6710 . . . . . . . . . . . . 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 258 . . . . . . . . . . 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 4817 . . . . . . . . . . 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 4164 . . . . . . . . . . . . . 14 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
106 iunin1 4887 . . . . . . . . . . . . . 14 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
107105, 106eqtr4i 2820 . . . . . . . . . . . . 13 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
108107imaeq2i 5796 . . . . . . . . . . . 12 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
109 imaiun 6860 . . . . . . . . . . . 12 (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
110108, 109eqtri 2817 . . . . . . . . . . 11 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
111 inss2 4121 . . . . . . . . . . . . . . . . . . . . 21 (ℂ ∩ ran 𝑔) ⊆ ran 𝑔
112 fnfvelrn 6704 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 Fn 𝐴𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
11385, 112sylan 580 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
114 intss1 4791 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔𝑥) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑥))
115113, 114syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ran 𝑔 ⊆ (𝑔𝑥))
116111, 115syl5ss 3895 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (ℂ ∩ ran 𝑔) ⊆ (𝑔𝑥))
117116ssdifd 4033 . . . . . . . . . . . . . . . . . . 19 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}))
118 sslin 4126 . . . . . . . . . . . . . . . . . . 19 (((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}) → (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
119 imass2 5833 . . . . . . . . . . . . . . . . . . 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 4164 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})) = (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))
122121imaeq2i 5796 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
123 inss1 4120 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) ⊆ 𝐵
124 resima2 5761 . . . . . . . . . . . . . . . . . . . 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 2817 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
127120, 126syl6sseqr 3934 . . . . . . . . . . . . . . . . 17 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
128 sstr2 3891 . . . . . . . . . . . . . . . . 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 3142 . . . . . . . . . . . . . 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 4862 . . . . . . . . . . . 12 ( 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢 ↔ ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
135133, 134sylibr 235 . . . . . . . . . . 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, 135syl5eqss 3931 . . . . . . . . . 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 2869 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐶𝑣𝐶 ∈ (ℂ ∩ ran 𝑔)))
138 ineq1 4096 . . . . . . . . . . . . . 14 (𝑣 = (ℂ ∩ ran 𝑔) → (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})))
139138imaeq2d 5798 . . . . . . . . . . . . 13 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))))
140139sseq1d 3914 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → ((𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
141137, 140anbi12d 630 . . . . . . . . . . 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 3554 . . . . . . . . . 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 833 . . . . . . . . 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 1911 . . . . . . . 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 3147 . . . . . 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 4862 . . . . . . . . 9 ( 𝑥𝐴 𝐵 ⊆ ℂ ↔ ∀𝑥𝐴 𝐵 ⊆ ℂ)
14935, 148sylibr 235 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ⊆ ℂ)
150149adantr 481 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑥𝐴 𝐵 ⊆ ℂ)
151147, 150, 94, 44ellimc2 24146 . . . . . 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 709 . . . . 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 243 . . 3 (𝜑 → (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
155154ssrdv 3890 . 2 (𝜑 → (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ⊆ (𝐹 lim 𝐶))
1567, 155eqssd 3901 1 (𝜑 → (𝐹 lim 𝐶) = (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1520  wex 1759  wcel 2079  wral 3103  wrex 3104  csb 3806  cdif 3851  cin 3853  wss 3854  {csn 4466   cint 4776   ciun 4819   ciin 4820  ran crn 5436  cres 5437  cima 5438   Fn wfn 6212  wf 6213  ontowfo 6215  cfv 6217  (class class class)co 7007  Fincfn 8347  cc 10370  TopOpenctopn 16512  fldccnfld 20215  Topctop 21173   lim climc 24131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-rep 5075  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214  ax-un 7310  ax-cnex 10428  ax-resscn 10429  ax-1cn 10430  ax-icn 10431  ax-addcl 10432  ax-addrcl 10433  ax-mulcl 10434  ax-mulrcl 10435  ax-mulcom 10436  ax-addass 10437  ax-mulass 10438  ax-distr 10439  ax-i2m1 10440  ax-1ne0 10441  ax-1rid 10442  ax-rnegex 10443  ax-rrecex 10444  ax-cnre 10445  ax-pre-lttri 10446  ax-pre-lttrn 10447  ax-pre-ltadd 10448  ax-pre-mulgt0 10449  ax-pre-sup 10450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-pss 3871  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-tp 4471  df-op 4473  df-uni 4740  df-int 4777  df-iun 4821  df-iin 4822  df-br 4957  df-opab 5019  df-mpt 5036  df-tr 5058  df-id 5340  df-eprel 5345  df-po 5354  df-so 5355  df-fr 5394  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-ord 6061  df-on 6062  df-lim 6063  df-suc 6064  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-f1 6222  df-fo 6223  df-f1o 6224  df-fv 6225  df-riota 6968  df-ov 7010  df-oprab 7011  df-mpo 7012  df-om 7428  df-1st 7536  df-2nd 7537  df-wrecs 7789  df-recs 7851  df-rdg 7889  df-1o 7944  df-oadd 7948  df-er 8130  df-map 8249  df-pm 8250  df-en 8348  df-dom 8349  df-sdom 8350  df-fin 8351  df-fi 8711  df-sup 8742  df-inf 8743  df-pnf 10512  df-mnf 10513  df-xr 10514  df-ltxr 10515  df-le 10516  df-sub 10708  df-neg 10709  df-div 11135  df-nn 11476  df-2 11537  df-3 11538  df-4 11539  df-5 11540  df-6 11541  df-7 11542  df-8 11543  df-9 11544  df-n0 11735  df-z 11819  df-dec 11937  df-uz 12083  df-q 12187  df-rp 12229  df-xneg 12346  df-xadd 12347  df-xmul 12348  df-fz 12732  df-seq 13208  df-exp 13268  df-cj 14280  df-re 14281  df-im 14282  df-sqrt 14416  df-abs 14417  df-struct 16302  df-ndx 16303  df-slot 16304  df-base 16306  df-plusg 16395  df-mulr 16396  df-starv 16397  df-tset 16401  df-ple 16402  df-ds 16404  df-unif 16405  df-rest 16513  df-topn 16514  df-topgen 16534  df-psmet 20207  df-xmet 20208  df-met 20209  df-bl 20210  df-mopn 20211  df-cnfld 20216  df-top 21174  df-topon 21191  df-topsp 21213  df-bases 21226  df-cnp 21508  df-xms 22601  df-ms 22602  df-limc 24135
This theorem is referenced by:  limcun  24164
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