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Theorem limciun 25797
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 25778 . . . 4 (𝐹 lim 𝐶) ⊆ ℂ
2 limcresi 25788 . . . . . 6 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
32rgenw 3060 . . . . 5 𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
4 ssiin 5052 . . . . 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 4227 . . 3 (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶))
76a1i 11 . 2 (𝜑 → (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
8 elriin 5078 . . . 4 (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ↔ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)))
9 simprl 770 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ ℂ)
10 limciun.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
1110ad2antrr 725 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → 𝐴 ∈ Fin)
12 simplrr 777 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))
13 nfcv 2898 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
14 nfcsb1v 3914 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎 / 𝑥𝐵
1513, 14nfres 5981 . . . . . . . . . . . . . . . . . . 19 𝑥(𝐹𝑎 / 𝑥𝐵)
16 nfcv 2898 . . . . . . . . . . . . . . . . . . 19 𝑥 lim
17 nfcv 2898 . . . . . . . . . . . . . . . . . . 19 𝑥𝐶
1815, 16, 17nfov 7444 . . . . . . . . . . . . . . . . . 18 𝑥((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
1918nfcri 2885 . . . . . . . . . . . . . . . . 17 𝑥 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
20 csbeq1a 3903 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
2120reseq2d 5979 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐹𝐵) = (𝐹𝑎 / 𝑥𝐵))
2221oveq1d 7429 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝐹𝐵) lim 𝐶) = ((𝐹𝑎 / 𝑥𝐵) lim 𝐶))
2322eleq2d 2814 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝑦 ∈ ((𝐹𝐵) lim 𝐶) ↔ 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)))
2419, 23rspc 3595 . . . . . . . . . . . . . . . 16 (𝑎𝐴 → (∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶) → 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)))
2512, 24mpan9 506 . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
28 ssiun2 5044 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝐴𝑎 / 𝑥𝐵 𝑎𝐴 𝑎 / 𝑥𝐵)
29 nfcv 2898 . . . . . . . . . . . . . . . . . . . . 21 𝑎𝐵
3029, 14, 20cbviun 5033 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐴 𝐵 = 𝑎𝐴 𝑎 / 𝑥𝐵
3128, 30sseqtrrdi 4029 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐴𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3231adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3327, 32fssresd 6758 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → (𝐹𝑎 / 𝑥𝐵):𝑎 / 𝑥𝐵⟶ℂ)
34 simpr 484 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎𝐴)
35 limciun.2 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑥𝐴 𝐵 ⊆ ℂ)
3635ad2antrr 725 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ⊆ ℂ)
37 nfcv 2898 . . . . . . . . . . . . . . . . . . . 20 𝑥
3814, 37nfss 3970 . . . . . . . . . . . . . . . . . . 19 𝑥𝑎 / 𝑥𝐵 ⊆ ℂ
3920sseq1d 4009 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐵 ⊆ ℂ ↔ 𝑎 / 𝑥𝐵 ⊆ ℂ))
4038, 39rspc 3595 . . . . . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐶 ∈ ℂ)
44 eqid 2727 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4533, 41, 43, 44ellimc2 25780 . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑢 ∈ (TopOpen‘ℂfld))
50 simplrr 777 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑦𝑢)
51 rsp 3239 . . . . . . . . . . . . 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 3141 . . . . . . . . . . 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 1910 . . . . . . . . . . . 12 𝑎𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)
55 nfcv 2898 . . . . . . . . . . . . 13 𝑥(TopOpen‘ℂfld)
56 nfv 1910 . . . . . . . . . . . . . 14 𝑥 𝐶𝑘
57 nfcv 2898 . . . . . . . . . . . . . . . . 17 𝑥𝑘
58 nfcv 2898 . . . . . . . . . . . . . . . . . 18 𝑥{𝐶}
5914, 58nfdif 4121 . . . . . . . . . . . . . . . . 17 𝑥(𝑎 / 𝑥𝐵 ∖ {𝐶})
6057, 59nfin 4212 . . . . . . . . . . . . . . . 16 𝑥(𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6115, 60nfima 6065 . . . . . . . . . . . . . . 15 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
62 nfcv 2898 . . . . . . . . . . . . . . 15 𝑥𝑢
6361, 62nfss 3970 . . . . . . . . . . . . . 14 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢
6456, 63nfan 1895 . . . . . . . . . . . . 13 𝑥(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6555, 64nfrexw 3305 . . . . . . . . . . . 12 𝑥𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6620difeq1d 4117 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝐵 ∖ {𝐶}) = (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6766ineq2d 4208 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑘 ∩ (𝐵 ∖ {𝐶})) = (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
6821, 67imaeq12d 6058 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))))
6968sseq1d 4009 . . . . . . . . . . . . . 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 3173 . . . . . . . . . . . 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 3298 . . . . . . . . . . 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 2817 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (𝐶𝑘𝐶 ∈ (𝑔𝑥)))
75 ineq1 4201 . . . . . . . . . . . . . 14 (𝑘 = (𝑔𝑥) → (𝑘 ∩ (𝐵 ∖ {𝐶})) = ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})))
7675imaeq2d 6057 . . . . . . . . . . . . 13 (𝑘 = (𝑔𝑥) → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
7776sseq1d 4009 . . . . . . . . . . . 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 9301 . . . . . . . . . 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 583 . . . . . . . . 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 24674 . . . . . . . . . . 11 (TopOpen‘ℂfld) ∈ Top
82 frn 6723 . . . . . . . . . . . 12 (𝑔:𝐴⟶(TopOpen‘ℂfld) → ran 𝑔 ⊆ (TopOpen‘ℂfld))
8382ad2antrl 727 . . . . . . . . . . 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 480 . . . . . . . . . . . 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 6716 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → 𝑔 Fn 𝐴)
8685ad2antrl 727 . . . . . . . . . . . . 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 6811 . . . . . . . . . . . . 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 9352 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑔:𝐴onto→ran 𝑔) → ran 𝑔 ∈ Fin)
9084, 88, 89syl2anc 583 . . . . . . . . . . 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 24676 . . . . . . . . . . . 12 ℂ = (TopOpen‘ℂfld)
9291rintopn 22785 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ ran 𝑔 ⊆ (TopOpen‘ℂfld) ∧ ran 𝑔 ∈ Fin) → (ℂ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld))
9381, 83, 90, 92mp3an2i 1463 . . . . . . . . . 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 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝐶 ∈ ℂ)
9594ad2antrr 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))) → 𝐶 ∈ ℂ)
96 simpl 482 . . . . . . . . . . . . . 14 ((𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → 𝐶 ∈ (𝑔𝑥))
9796ralimi 3078 . . . . . . . . . . . . 13 (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥))
9897ad2antll 728 . . . . . . . . . . . 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 2817 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑥) → (𝐶𝑧𝐶 ∈ (𝑔𝑥)))
10099ralrn 7092 . . . . . . . . . . . . 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 257 . . . . . . . . . . 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 4989 . . . . . . . . . . 11 (𝐶 ∈ (ℂ ∩ ran 𝑔) ↔ (𝐶 ∈ ℂ ∧ ∀𝑧 ∈ ran 𝑔 𝐶𝑧))
10495, 102, 103sylanbrc 582 . . . . . . . . . 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 4268 . . . . . . . . . . . . . 14 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
106 iunin1 5069 . . . . . . . . . . . . . 14 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
107105, 106eqtr4i 2758 . . . . . . . . . . . . 13 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
108107imaeq2i 6055 . . . . . . . . . . . 12 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
109 imaiun 7249 . . . . . . . . . . . 12 (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
110108, 109eqtri 2755 . . . . . . . . . . 11 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
111 inss2 4225 . . . . . . . . . . . . . . . . . . . . 21 (ℂ ∩ ran 𝑔) ⊆ ran 𝑔
112 fnfvelrn 7084 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 Fn 𝐴𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
11385, 112sylan 579 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
114 intss1 4961 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔𝑥) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑥))
115113, 114syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ran 𝑔 ⊆ (𝑔𝑥))
116111, 115sstrid 3989 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (ℂ ∩ ran 𝑔) ⊆ (𝑔𝑥))
117116ssdifd 4136 . . . . . . . . . . . . . . . . . . 19 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}))
118 sslin 4230 . . . . . . . . . . . . . . . . . . 19 (((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}) → (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
119 imass2 6100 . . . . . . . . . . . . . . . . . . 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 4268 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})) = (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))
122121imaeq2i 6055 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
123 inss1 4224 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) ⊆ 𝐵
124 resima2 6014 . . . . . . . . . . . . . . . . . . . 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 2755 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
127120, 126sseqtrrdi 4029 . . . . . . . . . . . . . . . . 17 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
128 sstr2 3985 . . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . 15 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
131130ralimdva 3162 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
132131imp 406 . . . . . . . . . . . . 13 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
133132adantl 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))) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
134 iunss 5042 . . . . . . . . . . . 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 4026 . . . . . . . . . 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 2817 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐶𝑣𝐶 ∈ (ℂ ∩ ran 𝑔)))
138 ineq1 4201 . . . . . . . . . . . . . 14 (𝑣 = (ℂ ∩ ran 𝑔) → (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})))
139138imaeq2d 6057 . . . . . . . . . . . . 13 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))))
140139sseq1d 4009 . . . . . . . . . . . 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 3607 . . . . . . . . . 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 836 . . . . . . . . 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 1931 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
145144expr 456 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
146145ralrimiva 3141 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
14726adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
148 iunss 5042 . . . . . . . . 9 ( 𝑥𝐴 𝐵 ⊆ ℂ ↔ ∀𝑥𝐴 𝐵 ⊆ ℂ)
14935, 148sylibr 233 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ⊆ ℂ)
150149adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑥𝐴 𝐵 ⊆ ℂ)
151147, 150, 94, 44ellimc2 25780 . . . . . 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 712 . . . . 5 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ (𝐹 lim 𝐶))
153152ex 412 . . . 4 (𝜑 → ((𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
1548, 153biimtrid 241 . . 3 (𝜑 → (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
155154ssrdv 3984 . 2 (𝜑 → (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ⊆ (𝐹 lim 𝐶))
1567, 155eqssd 3995 1 (𝜑 → (𝐹 lim 𝐶) = (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1534  wex 1774  wcel 2099  wral 3056  wrex 3065  csb 3889  cdif 3941  cin 3943  wss 3944  {csn 4624   cint 4944   ciun 4991   ciin 4992  ran crn 5673  cres 5674  cima 5675   Fn wfn 6537  wf 6538  ontowfo 6540  cfv 6542  (class class class)co 7414  Fincfn 8953  cc 11122  TopOpenctopn 17388  fldccnfld 21259  Topctop 22769   lim climc 25765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-tp 4629  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-iin 4994  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-pm 8837  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-fi 9420  df-sup 9451  df-inf 9452  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-4 12293  df-5 12294  df-6 12295  df-7 12296  df-8 12297  df-9 12298  df-n0 12489  df-z 12575  df-dec 12694  df-uz 12839  df-q 12949  df-rp 12993  df-xneg 13110  df-xadd 13111  df-xmul 13112  df-fz 13503  df-seq 13985  df-exp 14045  df-cj 15064  df-re 15065  df-im 15066  df-sqrt 15200  df-abs 15201  df-struct 17101  df-slot 17136  df-ndx 17148  df-base 17166  df-plusg 17231  df-mulr 17232  df-starv 17233  df-tset 17237  df-ple 17238  df-ds 17240  df-unif 17241  df-rest 17389  df-topn 17390  df-topgen 17410  df-psmet 21251  df-xmet 21252  df-met 21253  df-bl 21254  df-mopn 21255  df-cnfld 21260  df-top 22770  df-topon 22787  df-topsp 22809  df-bases 22823  df-cnp 23106  df-xms 24200  df-ms 24201  df-limc 25769
This theorem is referenced by:  limcun  25798
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