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Theorem limccl 13028
Description: Closure of the limit operator. (Contributed by Mario Carneiro, 25-Dec-2016.)
Assertion
Ref Expression
limccl (𝐹 lim 𝐵) ⊆ ℂ

Proof of Theorem limccl
Dummy variables 𝑑 𝑒 𝑓 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 id 19 . . . 4 (𝑤 ∈ (𝐹 lim 𝐵) → 𝑤 ∈ (𝐹 lim 𝐵))
2 df-limced 13025 . . . . . 6 lim = (𝑓 ∈ (ℂ ↑pm ℂ), 𝑥 ∈ ℂ ↦ {𝑦 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒)))})
32elmpocl1 6019 . . . . 5 (𝑤 ∈ (𝐹 lim 𝐵) → 𝐹 ∈ (ℂ ↑pm ℂ))
4 limcrcl 13027 . . . . . 6 (𝑤 ∈ (𝐹 lim 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
54simp3d 996 . . . . 5 (𝑤 ∈ (𝐹 lim 𝐵) → 𝐵 ∈ ℂ)
6 cnex 7856 . . . . . . 7 ℂ ∈ V
76rabex 4108 . . . . . 6 {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))} ∈ V
87a1i 9 . . . . 5 (𝑤 ∈ (𝐹 lim 𝐵) → {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))} ∈ V)
9 simpl 108 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → 𝑓 = 𝐹)
109dmeqd 4788 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → dom 𝑓 = dom 𝐹)
119, 10feq12d 5309 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
1210sseq1d 3157 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (dom 𝑓 ⊆ ℂ ↔ dom 𝐹 ⊆ ℂ))
1311, 12anbi12d 465 . . . . . . . 8 ((𝑓 = 𝐹𝑥 = 𝐵) → ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ)))
14 simpr 109 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → 𝑥 = 𝐵)
1514eleq1d 2226 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ))
1614breq2d 3977 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑧 # 𝑥𝑧 # 𝐵))
1714oveq2d 5840 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑧𝑥) = (𝑧𝐵))
1817fveq2d 5472 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹𝑥 = 𝐵) → (abs‘(𝑧𝑥)) = (abs‘(𝑧𝐵)))
1918breq1d 3975 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → ((abs‘(𝑧𝑥)) < 𝑑 ↔ (abs‘(𝑧𝐵)) < 𝑑))
2016, 19anbi12d 465 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑥 = 𝐵) → ((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) ↔ (𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑)))
219fveq1d 5470 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑓𝑧) = (𝐹𝑧))
2221fvoveq1d 5846 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → (abs‘((𝑓𝑧) − 𝑦)) = (abs‘((𝐹𝑧) − 𝑦)))
2322breq1d 3975 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑥 = 𝐵) → ((abs‘((𝑓𝑧) − 𝑦)) < 𝑒 ↔ (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))
2420, 23imbi12d 233 . . . . . . . . . . . 12 ((𝑓 = 𝐹𝑥 = 𝐵) → (((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒) ↔ ((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
2510, 24raleqbidv 2664 . . . . . . . . . . 11 ((𝑓 = 𝐹𝑥 = 𝐵) → (∀𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
2625rexbidv 2458 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → (∃𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒) ↔ ∃𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
2726ralbidv 2457 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒) ↔ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))
2815, 27anbi12d 465 . . . . . . . 8 ((𝑓 = 𝐹𝑥 = 𝐵) → ((𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒)) ↔ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒))))
2913, 28anbi12d 465 . . . . . . 7 ((𝑓 = 𝐹𝑥 = 𝐵) → (((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒))) ↔ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))))
3029rabbidv 2701 . . . . . 6 ((𝑓 = 𝐹𝑥 = 𝐵) → {𝑦 ∈ ℂ ∣ ((𝑓:dom 𝑓⟶ℂ ∧ dom 𝑓 ⊆ ℂ) ∧ (𝑥 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝑓((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) → (abs‘((𝑓𝑧) − 𝑦)) < 𝑒)))} = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))})
3130, 2ovmpoga 5950 . . . . 5 ((𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐵 ∈ ℂ ∧ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))} ∈ V) → (𝐹 lim 𝐵) = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))})
323, 5, 8, 31syl3anc 1220 . . . 4 (𝑤 ∈ (𝐹 lim 𝐵) → (𝐹 lim 𝐵) = {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))})
331, 32eleqtrd 2236 . . 3 (𝑤 ∈ (𝐹 lim 𝐵) → 𝑤 ∈ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))})
34 elrabi 2865 . . 3 (𝑤 ∈ {𝑦 ∈ ℂ ∣ ((𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ) ∧ (𝐵 ∈ ℂ ∧ ∀𝑒 ∈ ℝ+𝑑 ∈ ℝ+𝑧 ∈ dom 𝐹((𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑) → (abs‘((𝐹𝑧) − 𝑦)) < 𝑒)))} → 𝑤 ∈ ℂ)
3533, 34syl 14 . 2 (𝑤 ∈ (𝐹 lim 𝐵) → 𝑤 ∈ ℂ)
3635ssriv 3132 1 (𝐹 lim 𝐵) ⊆ ℂ
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1335  wcel 2128  wral 2435  wrex 2436  {crab 2439  Vcvv 2712  wss 3102   class class class wbr 3965  dom cdm 4586  wf 5166  cfv 5170  (class class class)co 5824  pm cpm 6594  cc 7730   < clt 7912  cmin 8046   # cap 8456  +crp 9560  abscabs 10897   lim climc 13023
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pm 6596  df-limced 13025
This theorem is referenced by:  reldvg  13048  dvfvalap  13050  dvcl  13052
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