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Theorem limcflf 23451
 Description: The limit operator can be expressed as a filter limit, from the filter of neighborhoods of 𝐵 restricted to 𝐴 ∖ {𝐵}, to the topology of the complex numbers. (If 𝐵 is not a limit point of 𝐴, then it is still formally a filter limit, but the neighborhood filter is not a proper filter in this case.) (Contributed by Mario Carneiro, 25-Dec-2016.)
Hypotheses
Ref Expression
limcflf.f (𝜑𝐹:𝐴⟶ℂ)
limcflf.a (𝜑𝐴 ⊆ ℂ)
limcflf.b (𝜑𝐵 ∈ ((limPt‘𝐾)‘𝐴))
limcflf.k 𝐾 = (TopOpen‘ℂfld)
limcflf.c 𝐶 = (𝐴 ∖ {𝐵})
limcflf.l 𝐿 = (((nei‘𝐾)‘{𝐵}) ↾t 𝐶)
Assertion
Ref Expression
limcflf (𝜑 → (𝐹 lim 𝐵) = ((𝐾 fLimf 𝐿)‘(𝐹𝐶)))

Proof of Theorem limcflf
Dummy variables 𝑡 𝑠 𝑢 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . . . . . . . . 11 𝑡 ∈ V
21inex1 4727 . . . . . . . . . 10 (𝑡𝐶) ∈ V
32rgenw 2908 . . . . . . . . 9 𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝑡𝐶) ∈ V
4 eqid 2610 . . . . . . . . . 10 (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↦ (𝑡𝐶)) = (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↦ (𝑡𝐶))
5 imaeq2 5381 . . . . . . . . . . . 12 (𝑠 = (𝑡𝐶) → ((𝐹𝐶) “ 𝑠) = ((𝐹𝐶) “ (𝑡𝐶)))
6 inss2 3796 . . . . . . . . . . . . 13 (𝑡𝐶) ⊆ 𝐶
7 resima2 5352 . . . . . . . . . . . . 13 ((𝑡𝐶) ⊆ 𝐶 → ((𝐹𝐶) “ (𝑡𝐶)) = (𝐹 “ (𝑡𝐶)))
86, 7ax-mp 5 . . . . . . . . . . . 12 ((𝐹𝐶) “ (𝑡𝐶)) = (𝐹 “ (𝑡𝐶))
95, 8syl6eq 2660 . . . . . . . . . . 11 (𝑠 = (𝑡𝐶) → ((𝐹𝐶) “ 𝑠) = (𝐹 “ (𝑡𝐶)))
109sseq1d 3595 . . . . . . . . . 10 (𝑠 = (𝑡𝐶) → (((𝐹𝐶) “ 𝑠) ⊆ 𝑢 ↔ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢))
114, 10rexrnmpt 6277 . . . . . . . . 9 (∀𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝑡𝐶) ∈ V → (∃𝑠 ∈ ran (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↦ (𝑡𝐶))((𝐹𝐶) “ 𝑠) ⊆ 𝑢 ↔ ∃𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝐹 “ (𝑡𝐶)) ⊆ 𝑢))
123, 11mp1i 13 . . . . . . . 8 (((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) → (∃𝑠 ∈ ran (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↦ (𝑡𝐶))((𝐹𝐶) “ 𝑠) ⊆ 𝑢 ↔ ∃𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝐹 “ (𝑡𝐶)) ⊆ 𝑢))
13 limcflf.l . . . . . . . . . 10 𝐿 = (((nei‘𝐾)‘{𝐵}) ↾t 𝐶)
14 fvex 6113 . . . . . . . . . . 11 ((nei‘𝐾)‘{𝐵}) ∈ V
15 limcflf.c . . . . . . . . . . . . . . 15 𝐶 = (𝐴 ∖ {𝐵})
16 difss 3699 . . . . . . . . . . . . . . 15 (𝐴 ∖ {𝐵}) ⊆ 𝐴
1715, 16eqsstri 3598 . . . . . . . . . . . . . 14 𝐶𝐴
18 limcflf.a . . . . . . . . . . . . . 14 (𝜑𝐴 ⊆ ℂ)
1917, 18syl5ss 3579 . . . . . . . . . . . . 13 (𝜑𝐶 ⊆ ℂ)
20 cnex 9896 . . . . . . . . . . . . . 14 ℂ ∈ V
2120ssex 4730 . . . . . . . . . . . . 13 (𝐶 ⊆ ℂ → 𝐶 ∈ V)
2219, 21syl 17 . . . . . . . . . . . 12 (𝜑𝐶 ∈ V)
2322ad2antrr 758 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) → 𝐶 ∈ V)
24 restval 15910 . . . . . . . . . . 11 ((((nei‘𝐾)‘{𝐵}) ∈ V ∧ 𝐶 ∈ V) → (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) = ran (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↦ (𝑡𝐶)))
2514, 23, 24sylancr 694 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) → (((nei‘𝐾)‘{𝐵}) ↾t 𝐶) = ran (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↦ (𝑡𝐶)))
2613, 25syl5eq 2656 . . . . . . . . 9 (((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) → 𝐿 = ran (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↦ (𝑡𝐶)))
2726rexeqdv 3122 . . . . . . . 8 (((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) → (∃𝑠𝐿 ((𝐹𝐶) “ 𝑠) ⊆ 𝑢 ↔ ∃𝑠 ∈ ran (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↦ (𝑡𝐶))((𝐹𝐶) “ 𝑠) ⊆ 𝑢))
28 limcflf.k . . . . . . . . . . . . . 14 𝐾 = (TopOpen‘ℂfld)
2928cnfldtop 22397 . . . . . . . . . . . . 13 𝐾 ∈ Top
30 opnneip 20733 . . . . . . . . . . . . 13 ((𝐾 ∈ Top ∧ 𝑤𝐾𝐵𝑤) → 𝑤 ∈ ((nei‘𝐾)‘{𝐵}))
3129, 30mp3an1 1403 . . . . . . . . . . . 12 ((𝑤𝐾𝐵𝑤) → 𝑤 ∈ ((nei‘𝐾)‘{𝐵}))
32 id 22 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑤𝑡 = 𝑤)
3315a1i 11 . . . . . . . . . . . . . . . 16 (𝑡 = 𝑤𝐶 = (𝐴 ∖ {𝐵}))
3432, 33ineq12d 3777 . . . . . . . . . . . . . . 15 (𝑡 = 𝑤 → (𝑡𝐶) = (𝑤 ∩ (𝐴 ∖ {𝐵})))
3534imaeq2d 5385 . . . . . . . . . . . . . 14 (𝑡 = 𝑤 → (𝐹 “ (𝑡𝐶)) = (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))))
3635sseq1d 3595 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → ((𝐹 “ (𝑡𝐶)) ⊆ 𝑢 ↔ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
3736rspcev 3282 . . . . . . . . . . . 12 ((𝑤 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) → ∃𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝐹 “ (𝑡𝐶)) ⊆ 𝑢)
3831, 37sylan 487 . . . . . . . . . . 11 (((𝑤𝐾𝐵𝑤) ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) → ∃𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝐹 “ (𝑡𝐶)) ⊆ 𝑢)
3938anasss 677 . . . . . . . . . 10 ((𝑤𝐾 ∧ (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) → ∃𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝐹 “ (𝑡𝐶)) ⊆ 𝑢)
4039rexlimiva 3010 . . . . . . . . 9 (∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) → ∃𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝐹 “ (𝑡𝐶)) ⊆ 𝑢)
41 simprl 790 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → 𝑡 ∈ ((nei‘𝐾)‘{𝐵}))
4228cnfldtopon 22396 . . . . . . . . . . . . . . 15 𝐾 ∈ (TopOn‘ℂ)
4342toponunii 20547 . . . . . . . . . . . . . 14 ℂ = 𝐾
4443neii1 20720 . . . . . . . . . . . . 13 ((𝐾 ∈ Top ∧ 𝑡 ∈ ((nei‘𝐾)‘{𝐵})) → 𝑡 ⊆ ℂ)
4529, 41, 44sylancr 694 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → 𝑡 ⊆ ℂ)
4643ntropn 20663 . . . . . . . . . . . 12 ((𝐾 ∈ Top ∧ 𝑡 ⊆ ℂ) → ((int‘𝐾)‘𝑡) ∈ 𝐾)
4729, 45, 46sylancr 694 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → ((int‘𝐾)‘𝑡) ∈ 𝐾)
4829a1i 11 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → 𝐾 ∈ Top)
4943lpss 20756 . . . . . . . . . . . . . . . . . 18 ((𝐾 ∈ Top ∧ 𝐴 ⊆ ℂ) → ((limPt‘𝐾)‘𝐴) ⊆ ℂ)
5029, 18, 49sylancr 694 . . . . . . . . . . . . . . . . 17 (𝜑 → ((limPt‘𝐾)‘𝐴) ⊆ ℂ)
51 limcflf.b . . . . . . . . . . . . . . . . 17 (𝜑𝐵 ∈ ((limPt‘𝐾)‘𝐴))
5250, 51sseldd 3569 . . . . . . . . . . . . . . . 16 (𝜑𝐵 ∈ ℂ)
5352snssd 4281 . . . . . . . . . . . . . . 15 (𝜑 → {𝐵} ⊆ ℂ)
5453ad3antrrr 762 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → {𝐵} ⊆ ℂ)
5543neiint 20718 . . . . . . . . . . . . . 14 ((𝐾 ∈ Top ∧ {𝐵} ⊆ ℂ ∧ 𝑡 ⊆ ℂ) → (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↔ {𝐵} ⊆ ((int‘𝐾)‘𝑡)))
5648, 54, 45, 55syl3anc 1318 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ↔ {𝐵} ⊆ ((int‘𝐾)‘𝑡)))
5741, 56mpbid 221 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → {𝐵} ⊆ ((int‘𝐾)‘𝑡))
5852ad3antrrr 762 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → 𝐵 ∈ ℂ)
59 snssg 4268 . . . . . . . . . . . . 13 (𝐵 ∈ ℂ → (𝐵 ∈ ((int‘𝐾)‘𝑡) ↔ {𝐵} ⊆ ((int‘𝐾)‘𝑡)))
6058, 59syl 17 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → (𝐵 ∈ ((int‘𝐾)‘𝑡) ↔ {𝐵} ⊆ ((int‘𝐾)‘𝑡)))
6157, 60mpbird 246 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → 𝐵 ∈ ((int‘𝐾)‘𝑡))
6243ntrss2 20671 . . . . . . . . . . . . . 14 ((𝐾 ∈ Top ∧ 𝑡 ⊆ ℂ) → ((int‘𝐾)‘𝑡) ⊆ 𝑡)
6329, 45, 62sylancr 694 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → ((int‘𝐾)‘𝑡) ⊆ 𝑡)
64 ssrin 3800 . . . . . . . . . . . . 13 (((int‘𝐾)‘𝑡) ⊆ 𝑡 → (((int‘𝐾)‘𝑡) ∩ 𝐶) ⊆ (𝑡𝐶))
65 imass2 5420 . . . . . . . . . . . . 13 ((((int‘𝐾)‘𝑡) ∩ 𝐶) ⊆ (𝑡𝐶) → (𝐹 “ (((int‘𝐾)‘𝑡) ∩ 𝐶)) ⊆ (𝐹 “ (𝑡𝐶)))
6663, 64, 653syl 18 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → (𝐹 “ (((int‘𝐾)‘𝑡) ∩ 𝐶)) ⊆ (𝐹 “ (𝑡𝐶)))
67 simprr 792 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)
6866, 67sstrd 3578 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → (𝐹 “ (((int‘𝐾)‘𝑡) ∩ 𝐶)) ⊆ 𝑢)
69 eleq2 2677 . . . . . . . . . . . . 13 (𝑤 = ((int‘𝐾)‘𝑡) → (𝐵𝑤𝐵 ∈ ((int‘𝐾)‘𝑡)))
7015ineq2i 3773 . . . . . . . . . . . . . . . 16 (𝑤𝐶) = (𝑤 ∩ (𝐴 ∖ {𝐵}))
71 ineq1 3769 . . . . . . . . . . . . . . . 16 (𝑤 = ((int‘𝐾)‘𝑡) → (𝑤𝐶) = (((int‘𝐾)‘𝑡) ∩ 𝐶))
7270, 71syl5eqr 2658 . . . . . . . . . . . . . . 15 (𝑤 = ((int‘𝐾)‘𝑡) → (𝑤 ∩ (𝐴 ∖ {𝐵})) = (((int‘𝐾)‘𝑡) ∩ 𝐶))
7372imaeq2d 5385 . . . . . . . . . . . . . 14 (𝑤 = ((int‘𝐾)‘𝑡) → (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) = (𝐹 “ (((int‘𝐾)‘𝑡) ∩ 𝐶)))
7473sseq1d 3595 . . . . . . . . . . . . 13 (𝑤 = ((int‘𝐾)‘𝑡) → ((𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢 ↔ (𝐹 “ (((int‘𝐾)‘𝑡) ∩ 𝐶)) ⊆ 𝑢))
7569, 74anbi12d 743 . . . . . . . . . . . 12 (𝑤 = ((int‘𝐾)‘𝑡) → ((𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ (𝐵 ∈ ((int‘𝐾)‘𝑡) ∧ (𝐹 “ (((int‘𝐾)‘𝑡) ∩ 𝐶)) ⊆ 𝑢)))
7675rspcev 3282 . . . . . . . . . . 11 ((((int‘𝐾)‘𝑡) ∈ 𝐾 ∧ (𝐵 ∈ ((int‘𝐾)‘𝑡) ∧ (𝐹 “ (((int‘𝐾)‘𝑡) ∩ 𝐶)) ⊆ 𝑢)) → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
7747, 61, 68, 76syl12anc 1316 . . . . . . . . . 10 ((((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) ∧ (𝑡 ∈ ((nei‘𝐾)‘{𝐵}) ∧ (𝐹 “ (𝑡𝐶)) ⊆ 𝑢)) → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))
7877rexlimdvaa 3014 . . . . . . . . 9 (((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) → (∃𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝐹 “ (𝑡𝐶)) ⊆ 𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))
7940, 78impbid2 215 . . . . . . . 8 (((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) → (∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ ∃𝑡 ∈ ((nei‘𝐾)‘{𝐵})(𝐹 “ (𝑡𝐶)) ⊆ 𝑢))
8012, 27, 793bitr4rd 300 . . . . . . 7 (((𝜑𝑥 ∈ ℂ) ∧ (𝑢𝐾𝑥𝑢)) → (∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ ∃𝑠𝐿 ((𝐹𝐶) “ 𝑠) ⊆ 𝑢))
8180anassrs 678 . . . . . 6 ((((𝜑𝑥 ∈ ℂ) ∧ 𝑢𝐾) ∧ 𝑥𝑢) → (∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢) ↔ ∃𝑠𝐿 ((𝐹𝐶) “ 𝑠) ⊆ 𝑢))
8281pm5.74da 719 . . . . 5 (((𝜑𝑥 ∈ ℂ) ∧ 𝑢𝐾) → ((𝑥𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) ↔ (𝑥𝑢 → ∃𝑠𝐿 ((𝐹𝐶) “ 𝑠) ⊆ 𝑢)))
8382ralbidva 2968 . . . 4 ((𝜑𝑥 ∈ ℂ) → (∀𝑢𝐾 (𝑥𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)) ↔ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑠𝐿 ((𝐹𝐶) “ 𝑠) ⊆ 𝑢)))
8483pm5.32da 671 . . 3 (𝜑 → ((𝑥 ∈ ℂ ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢))) ↔ (𝑥 ∈ ℂ ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑠𝐿 ((𝐹𝐶) “ 𝑠) ⊆ 𝑢))))
85 limcflf.f . . . 4 (𝜑𝐹:𝐴⟶ℂ)
8685, 18, 52, 28ellimc2 23447 . . 3 (𝜑 → (𝑥 ∈ (𝐹 lim 𝐵) ↔ (𝑥 ∈ ℂ ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑤𝐾 (𝐵𝑤 ∧ (𝐹 “ (𝑤 ∩ (𝐴 ∖ {𝐵}))) ⊆ 𝑢)))))
8742a1i 11 . . . 4 (𝜑𝐾 ∈ (TopOn‘ℂ))
8885, 18, 51, 28, 15, 13limcflflem 23450 . . . 4 (𝜑𝐿 ∈ (Fil‘𝐶))
89 fssres 5983 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐶𝐴) → (𝐹𝐶):𝐶⟶ℂ)
9085, 17, 89sylancl 693 . . . 4 (𝜑 → (𝐹𝐶):𝐶⟶ℂ)
91 isflf 21607 . . . 4 ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝐿 ∈ (Fil‘𝐶) ∧ (𝐹𝐶):𝐶⟶ℂ) → (𝑥 ∈ ((𝐾 fLimf 𝐿)‘(𝐹𝐶)) ↔ (𝑥 ∈ ℂ ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑠𝐿 ((𝐹𝐶) “ 𝑠) ⊆ 𝑢))))
9287, 88, 90, 91syl3anc 1318 . . 3 (𝜑 → (𝑥 ∈ ((𝐾 fLimf 𝐿)‘(𝐹𝐶)) ↔ (𝑥 ∈ ℂ ∧ ∀𝑢𝐾 (𝑥𝑢 → ∃𝑠𝐿 ((𝐹𝐶) “ 𝑠) ⊆ 𝑢))))
9384, 86, 923bitr4d 299 . 2 (𝜑 → (𝑥 ∈ (𝐹 lim 𝐵) ↔ 𝑥 ∈ ((𝐾 fLimf 𝐿)‘(𝐹𝐶))))
9493eqrdv 2608 1 (𝜑 → (𝐹 lim 𝐵) = ((𝐾 fLimf 𝐿)‘(𝐹𝐶)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  {csn 4125   ↦ cmpt 4643  ran crn 5039   ↾ cres 5040   “ cima 5041  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℂcc 9813   ↾t crest 15904  TopOpenctopn 15905  ℂfldccnfld 19567  Topctop 20517  TopOnctopon 20518  intcnt 20631  neicnei 20711  limPtclp 20748  Filcfil 21459   fLimf cflf 21549   limℂ climc 23432 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-fz 12198  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-plusg 15781  df-mulr 15782  df-starv 15783  df-tset 15787  df-ple 15788  df-ds 15791  df-unif 15792  df-rest 15906  df-topn 15907  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-cnfld 19568  df-top 20521  df-bases 20522  df-topon 20523  df-topsp 20524  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-lp 20750  df-cnp 20842  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-xms 21935  df-ms 21936  df-limc 23436 This theorem is referenced by:  limcmo  23452
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