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Theorem limccl 13422
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 13419 . . . . . 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 6048 . . . . 5 (𝑤 ∈ (𝐹 lim 𝐵) → 𝐹 ∈ (ℂ ↑pm ℂ))
4 limcrcl 13421 . . . . . 6 (𝑤 ∈ (𝐹 lim 𝐵) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ ∧ 𝐵 ∈ ℂ))
54simp3d 1006 . . . . 5 (𝑤 ∈ (𝐹 lim 𝐵) → 𝐵 ∈ ℂ)
6 cnex 7898 . . . . . . 7 ℂ ∈ V
76rabex 4133 . . . . . 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 4813 . . . . . . . . . 10 ((𝑓 = 𝐹𝑥 = 𝐵) → dom 𝑓 = dom 𝐹)
119, 10feq12d 5337 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑓:dom 𝑓⟶ℂ ↔ 𝐹:dom 𝐹⟶ℂ))
1210sseq1d 3176 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (dom 𝑓 ⊆ ℂ ↔ dom 𝐹 ⊆ ℂ))
1311, 12anbi12d 470 . . . . . . . 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 2239 . . . . . . . . 9 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑥 ∈ ℂ ↔ 𝐵 ∈ ℂ))
1614breq2d 4001 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑧 # 𝑥𝑧 # 𝐵))
1714oveq2d 5869 . . . . . . . . . . . . . . . 16 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑧𝑥) = (𝑧𝐵))
1817fveq2d 5500 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹𝑥 = 𝐵) → (abs‘(𝑧𝑥)) = (abs‘(𝑧𝐵)))
1918breq1d 3999 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → ((abs‘(𝑧𝑥)) < 𝑑 ↔ (abs‘(𝑧𝐵)) < 𝑑))
2016, 19anbi12d 470 . . . . . . . . . . . . 13 ((𝑓 = 𝐹𝑥 = 𝐵) → ((𝑧 # 𝑥 ∧ (abs‘(𝑧𝑥)) < 𝑑) ↔ (𝑧 # 𝐵 ∧ (abs‘(𝑧𝐵)) < 𝑑)))
219fveq1d 5498 . . . . . . . . . . . . . . 15 ((𝑓 = 𝐹𝑥 = 𝐵) → (𝑓𝑧) = (𝐹𝑧))
2221fvoveq1d 5875 . . . . . . . . . . . . . 14 ((𝑓 = 𝐹𝑥 = 𝐵) → (abs‘((𝑓𝑧) − 𝑦)) = (abs‘((𝐹𝑧) − 𝑦)))
2322breq1d 3999 . . . . . . . . . . . . 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 2677 . . . . . . . . . . 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 2471 . . . . . . . . . 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 2470 . . . . . . . . 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 470 . . . . . . . 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 470 . . . . . . 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 2719 . . . . . 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 5982 . . . . 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 1233 . . . 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 2249 . . 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 2883 . . 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 3151 1 (𝐹 lim 𝐵) ⊆ ℂ
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1348  wcel 2141  wral 2448  wrex 2449  {crab 2452  Vcvv 2730  wss 3121   class class class wbr 3989  dom cdm 4611  wf 5194  cfv 5198  (class class class)co 5853  pm cpm 6627  cc 7772   < clt 7954  cmin 8090   # cap 8500  +crp 9610  abscabs 10961   lim climc 13417
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pm 6629  df-limced 13419
This theorem is referenced by:  reldvg  13442  dvfvalap  13444  dvcl  13446
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