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

Theorem limciun 23703
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 23684 . . . 4 (𝐹 lim 𝐶) ⊆ ℂ
2 limcresi 23694 . . . . . 6 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
32rgenw 2953 . . . . 5 𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶)
4 ssiin 4602 . . . . 5 ((𝐹 lim 𝐶) ⊆ 𝑥𝐴 ((𝐹𝐵) lim 𝐶) ↔ ∀𝑥𝐴 (𝐹 lim 𝐶) ⊆ ((𝐹𝐵) lim 𝐶))
53, 4mpbir 221 . . . 4 (𝐹 lim 𝐶) ⊆ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)
61, 5ssini 3869 . . 3 (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶))
76a1i 11 . 2 (𝜑 → (𝐹 lim 𝐶) ⊆ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
8 elriin 4625 . . . 4 (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ↔ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)))
9 simprl 809 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ ℂ)
10 limciun.1 . . . . . . . . . . 11 (𝜑𝐴 ∈ Fin)
1110ad2antrr 762 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → 𝐴 ∈ Fin)
12 simplrr 818 . . . . . . . . . . . . . . . 16 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))
13 nfcv 2793 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐹
14 nfcsb1v 3582 . . . . . . . . . . . . . . . . . . . 20 𝑥𝑎 / 𝑥𝐵
1513, 14nfres 5430 . . . . . . . . . . . . . . . . . . 19 𝑥(𝐹𝑎 / 𝑥𝐵)
16 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑥 lim
17 nfcv 2793 . . . . . . . . . . . . . . . . . . 19 𝑥𝐶
1815, 16, 17nfov 6716 . . . . . . . . . . . . . . . . . 18 𝑥((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
1918nfcri 2787 . . . . . . . . . . . . . . . . 17 𝑥 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)
20 csbeq1a 3575 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎𝐵 = 𝑎 / 𝑥𝐵)
2120reseq2d 5428 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐹𝐵) = (𝐹𝑎 / 𝑥𝐵))
2221oveq1d 6705 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑎 → ((𝐹𝐵) lim 𝐶) = ((𝐹𝑎 / 𝑥𝐵) lim 𝐶))
2322eleq2d 2716 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝑦 ∈ ((𝐹𝐵) lim 𝐶) ↔ 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)))
2419, 23rspc 3334 . . . . . . . . . . . . . . . 16 (𝑎𝐴 → (∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶) → 𝑦 ∈ ((𝐹𝑎 / 𝑥𝐵) lim 𝐶)))
2512, 24mpan9 485 . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
28 ssiun2 4595 . . . . . . . . . . . . . . . . . . . 20 (𝑎𝐴𝑎 / 𝑥𝐵 𝑎𝐴 𝑎 / 𝑥𝐵)
29 nfcv 2793 . . . . . . . . . . . . . . . . . . . . 21 𝑎𝐵
3029, 14, 20cbviun 4589 . . . . . . . . . . . . . . . . . . . 20 𝑥𝐴 𝐵 = 𝑎𝐴 𝑎 / 𝑥𝐵
3128, 30syl6sseqr 3685 . . . . . . . . . . . . . . . . . . 19 (𝑎𝐴𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3231adantl 481 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎 / 𝑥𝐵 𝑥𝐴 𝐵)
3327, 32fssresd 6109 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → (𝐹𝑎 / 𝑥𝐵):𝑎 / 𝑥𝐵⟶ℂ)
34 simpr 476 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝑎𝐴)
35 limciun.2 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ∀𝑥𝐴 𝐵 ⊆ ℂ)
3635ad2antrr 762 . . . . . . . . . . . . . . . . . 18 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → ∀𝑥𝐴 𝐵 ⊆ ℂ)
37 nfcv 2793 . . . . . . . . . . . . . . . . . . . 20 𝑥
3814, 37nfss 3629 . . . . . . . . . . . . . . . . . . 19 𝑥𝑎 / 𝑥𝐵 ⊆ ℂ
3920sseq1d 3665 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑎 → (𝐵 ⊆ ℂ ↔ 𝑎 / 𝑥𝐵 ⊆ ℂ))
4038, 39rspc 3334 . . . . . . . . . . . . . . . . . 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 762 . . . . . . . . . . . . . . . . 17 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑎𝐴) → 𝐶 ∈ ℂ)
44 eqid 2651 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
4533, 41, 43, 44ellimc2 23686 . . . . . . . . . . . . . . . 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 751 . . . . . . . . . . . . . . 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 222 . . . . . . . . . . . . . 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 478 . . . . . . . . . . . . 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 817 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑢 ∈ (TopOpen‘ℂfld))
50 simplrr 818 . . . . . . . . . . . . 13 ((((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) ∧ 𝑎𝐴) → 𝑦𝑢)
51 rsp 2958 . . . . . . . . . . . . 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 2995 . . . . . . . . . . 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 1883 . . . . . . . . . . . 12 𝑎𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)
55 nfcv 2793 . . . . . . . . . . . . 13 𝑥(TopOpen‘ℂfld)
56 nfv 1883 . . . . . . . . . . . . . 14 𝑥 𝐶𝑘
57 nfcv 2793 . . . . . . . . . . . . . . . . 17 𝑥𝑘
58 nfcv 2793 . . . . . . . . . . . . . . . . . 18 𝑥{𝐶}
5914, 58nfdif 3764 . . . . . . . . . . . . . . . . 17 𝑥(𝑎 / 𝑥𝐵 ∖ {𝐶})
6057, 59nfin 3853 . . . . . . . . . . . . . . . 16 𝑥(𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6115, 60nfima 5509 . . . . . . . . . . . . . . 15 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
62 nfcv 2793 . . . . . . . . . . . . . . 15 𝑥𝑢
6361, 62nfss 3629 . . . . . . . . . . . . . 14 𝑥((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢
6456, 63nfan 1868 . . . . . . . . . . . . 13 𝑥(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6555, 64nfrex 3036 . . . . . . . . . . . 12 𝑥𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)
6620difeq1d 3760 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑎 → (𝐵 ∖ {𝐶}) = (𝑎 / 𝑥𝐵 ∖ {𝐶}))
6766ineq2d 3847 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑎 → (𝑘 ∩ (𝐵 ∖ {𝐶})) = (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶})))
6821, 67imaeq12d 5502 . . . . . . . . . . . . . . 15 (𝑥 = 𝑎 → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))))
6968sseq1d 3665 . . . . . . . . . . . . . 14 (𝑥 = 𝑎 → (((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7069anbi2d 740 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → ((𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7170rexbidv 3081 . . . . . . . . . . . 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 3197 . . . . . . . . . . 11 (∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ ∀𝑎𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝑎 / 𝑥𝐵) “ (𝑘 ∩ (𝑎 / 𝑥𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7353, 72sylibr 224 . . . . . . . . . 10 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∀𝑥𝐴𝑘 ∈ (TopOpen‘ℂfld)(𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
74 eleq2 2719 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (𝐶𝑘𝐶 ∈ (𝑔𝑥)))
75 ineq1 3840 . . . . . . . . . . . . . 14 (𝑘 = (𝑔𝑥) → (𝑘 ∩ (𝐵 ∖ {𝐶})) = ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})))
7675imaeq2d 5501 . . . . . . . . . . . . 13 (𝑘 = (𝑔𝑥) → ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
7776sseq1d 3665 . . . . . . . . . . . 12 (𝑘 = (𝑔𝑥) → (((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢))
7874, 77anbi12d 747 . . . . . . . . . . 11 (𝑘 = (𝑔𝑥) → ((𝐶𝑘 ∧ ((𝐹𝐵) “ (𝑘 ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) ↔ (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
7978ac6sfi 8245 . . . . . . . . . 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 694 . . . . . . . . 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 22634 . . . . . . . . . . . 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 6091 . . . . . . . . . . . 12 (𝑔:𝐴⟶(TopOpen‘ℂfld) → ran 𝑔 ⊆ (TopOpen‘ℂfld))
8483ad2antrl 764 . . . . . . . . . . 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 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)
86 ffn 6083 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → 𝑔 Fn 𝐴)
8786ad2antrl 764 . . . . . . . . . . . . 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 6159 . . . . . . . . . . . . 13 (𝑔 Fn 𝐴𝑔:𝐴onto→ran 𝑔)
8987, 88sylib 208 . . . . . . . . . . . 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 8293 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ 𝑔:𝐴onto→ran 𝑔) → ran 𝑔 ∈ Fin)
9185, 89, 90syl2anc 694 . . . . . . . . . . 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 22633 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
9392toponunii 20769 . . . . . . . . . . . 12 ℂ = (TopOpen‘ℂfld)
9493rintopn 20762 . . . . . . . . . . 11 (((TopOpen‘ℂfld) ∈ Top ∧ ran 𝑔 ⊆ (TopOpen‘ℂfld) ∧ ran 𝑔 ∈ Fin) → (ℂ ∩ ran 𝑔) ∈ (TopOpen‘ℂfld))
9582, 84, 91, 94syl3anc 1366 . . . . . . . . . 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 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝐶 ∈ ℂ)
9796ad2antrr 762 . . . . . . . . . . 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 472 . . . . . . . . . . . . . 14 ((𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → 𝐶 ∈ (𝑔𝑥))
9998ralimi 2981 . . . . . . . . . . . . 13 (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 𝐶 ∈ (𝑔𝑥))
10099ad2antll 765 . . . . . . . . . . . 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 2719 . . . . . . . . . . . . . 14 (𝑧 = (𝑔𝑥) → (𝐶𝑧𝐶 ∈ (𝑔𝑥)))
102101ralrn 6402 . . . . . . . . . . . . 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 247 . . . . . . . . . . 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 4550 . . . . . . . . . . 11 (𝐶 ∈ (ℂ ∩ ran 𝑔) ↔ (𝐶 ∈ ℂ ∧ ∀𝑧 ∈ ran 𝑔 𝐶𝑧))
10697, 104, 105sylanbrc 699 . . . . . . . . . 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 3905 . . . . . . . . . . . . . 14 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
108 iunin1 4617 . . . . . . . . . . . . . 14 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) = ( 𝑥𝐴 𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
109107, 108eqtr4i 2676 . . . . . . . . . . . . 13 ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))
110109imaeq2i 5499 . . . . . . . . . . . 12 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
111 imaiun 6543 . . . . . . . . . . . 12 (𝐹 𝑥𝐴 (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
112110, 111eqtri 2673 . . . . . . . . . . 11 (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})))
113 inss2 3867 . . . . . . . . . . . . . . . . . . . . 21 (ℂ ∩ ran 𝑔) ⊆ ran 𝑔
114 fnfvelrn 6396 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑔 Fn 𝐴𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
11586, 114sylan 487 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝑔𝑥) ∈ ran 𝑔)
116 intss1 4524 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑔𝑥) ∈ ran 𝑔 ran 𝑔 ⊆ (𝑔𝑥))
117115, 116syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ran 𝑔 ⊆ (𝑔𝑥))
118113, 117syl5ss 3647 . . . . . . . . . . . . . . . . . . . 20 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (ℂ ∩ ran 𝑔) ⊆ (𝑔𝑥))
119118ssdifd 3779 . . . . . . . . . . . . . . . . . . 19 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}))
120 sslin 3872 . . . . . . . . . . . . . . . . . . 19 (((ℂ ∩ ran 𝑔) ∖ {𝐶}) ⊆ ((𝑔𝑥) ∖ {𝐶}) → (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶})) ⊆ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
121 imass2 5536 . . . . . . . . . . . . . . . . . . 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 3905 . . . . . . . . . . . . . . . . . . . 20 ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶})) = (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶}))
124123imaeq2i 5499 . . . . . . . . . . . . . . . . . . 19 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = ((𝐹𝐵) “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
125 inss1 3866 . . . . . . . . . . . . . . . . . . . 20 (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})) ⊆ 𝐵
126 resima2 5467 . . . . . . . . . . . . . . . . . . . 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 2673 . . . . . . . . . . . . . . . . . 18 ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) = (𝐹 “ (𝐵 ∩ ((𝑔𝑥) ∖ {𝐶})))
129122, 128syl6sseqr 3685 . . . . . . . . . . . . . . . . 17 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))))
130 sstr2 3643 . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . 15 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ 𝑥𝐴) → ((𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
133132ralimdva 2991 . . . . . . . . . . . . . 14 (𝑔:𝐴⟶(TopOpen‘ℂfld) → (∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢))
134133imp 444 . . . . . . . . . . . . 13 ((𝑔:𝐴⟶(TopOpen‘ℂfld) ∧ ∀𝑥𝐴 (𝐶 ∈ (𝑔𝑥) ∧ ((𝐹𝐵) “ ((𝑔𝑥) ∩ (𝐵 ∖ {𝐶}))) ⊆ 𝑢)) → ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
135134adantl 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)
136 iunss 4593 . . . . . . . . . . . 12 ( 𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢 ↔ ∀𝑥𝐴 (𝐹 “ (𝐵 ∩ ((ℂ ∩ ran 𝑔) ∖ {𝐶}))) ⊆ 𝑢)
137135, 136sylibr 224 . . . . . . . . . . 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 3682 . . . . . . . . . 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 2719 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐶𝑣𝐶 ∈ (ℂ ∩ ran 𝑔)))
140 ineq1 3840 . . . . . . . . . . . . . 14 (𝑣 = (ℂ ∩ ran 𝑔) → (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})) = ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶})))
141140imaeq2d 5501 . . . . . . . . . . . . 13 (𝑣 = (ℂ ∩ ran 𝑔) → (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) = (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))))
142141sseq1d 3665 . . . . . . . . . . . 12 (𝑣 = (ℂ ∩ ran 𝑔) → ((𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢 ↔ (𝐹 “ ((ℂ ∩ ran 𝑔) ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
143139, 142anbi12d 747 . . . . . . . . . . 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 3340 . . . . . . . . . 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 1364 . . . . . . . . 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 1903 . . . . . . . 8 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ (𝑢 ∈ (TopOpen‘ℂfld) ∧ 𝑦𝑢)) → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢))
147146expr 642 . . . . . . 7 (((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) ∧ 𝑢 ∈ (TopOpen‘ℂfld)) → (𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
148147ralrimiva 2995 . . . . . 6 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → ∀𝑢 ∈ (TopOpen‘ℂfld)(𝑦𝑢 → ∃𝑣 ∈ (TopOpen‘ℂfld)(𝐶𝑣 ∧ (𝐹 “ (𝑣 ∩ ( 𝑥𝐴 𝐵 ∖ {𝐶}))) ⊆ 𝑢)))
14926adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝐹: 𝑥𝐴 𝐵⟶ℂ)
150 iunss 4593 . . . . . . . . 9 ( 𝑥𝐴 𝐵 ⊆ ℂ ↔ ∀𝑥𝐴 𝐵 ⊆ ℂ)
15135, 150sylibr 224 . . . . . . . 8 (𝜑 𝑥𝐴 𝐵 ⊆ ℂ)
152151adantr 480 . . . . . . 7 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑥𝐴 𝐵 ⊆ ℂ)
153149, 152, 96, 44ellimc2 23686 . . . . . 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 977 . . . . 5 ((𝜑 ∧ (𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶))) → 𝑦 ∈ (𝐹 lim 𝐶))
155154ex 449 . . . 4 (𝜑 → ((𝑦 ∈ ℂ ∧ ∀𝑥𝐴 𝑦 ∈ ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
1568, 155syl5bi 232 . . 3 (𝜑 → (𝑦 ∈ (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) → 𝑦 ∈ (𝐹 lim 𝐶)))
157156ssrdv 3642 . 2 (𝜑 → (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)) ⊆ (𝐹 lim 𝐶))
1587, 157eqssd 3653 1 (𝜑 → (𝐹 lim 𝐶) = (ℂ ∩ 𝑥𝐴 ((𝐹𝐵) lim 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wral 2941  wrex 2942  csb 3566  cdif 3604  cin 3606  wss 3607  {csn 4210   cint 4507   ciun 4552   ciin 4553  ran crn 5144  cres 5145  cima 5146   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  Fincfn 7997  cc 9972  TopOpenctopn 16129  fldccnfld 19794  Topctop 20746   lim climc 23671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fi 8358  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-fz 12365  df-seq 12842  df-exp 12901  df-cj 13883  df-re 13884  df-im 13885  df-sqrt 14019  df-abs 14020  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-plusg 16001  df-mulr 16002  df-starv 16003  df-tset 16007  df-ple 16008  df-ds 16011  df-unif 16012  df-rest 16130  df-topn 16131  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-cnfld 19795  df-top 20747  df-topon 20764  df-topsp 20785  df-bases 20798  df-cnp 21080  df-xms 22172  df-ms 22173  df-limc 23675
This theorem is referenced by:  limcun  23704
  Copyright terms: Public domain W3C validator