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Theorem elcncf 24915
Description: Membership in the set of continuous complex functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
Assertion
Ref Expression
elcncf ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐴   𝑤,𝐹,𝑥,𝑦,𝑧   𝑤,𝐵,𝑥,𝑦,𝑧

Proof of Theorem elcncf
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 cncfval 24914 . . . 4 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21eleq2d 2827 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ 𝐹 ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)}))
3 fveq1 6905 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
4 fveq1 6905 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑤) = (𝐹𝑤))
53, 4oveq12d 7449 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) − (𝑓𝑤)) = ((𝐹𝑥) − (𝐹𝑤)))
65fveq2d 6910 . . . . . . . 8 (𝑓 = 𝐹 → (abs‘((𝑓𝑥) − (𝑓𝑤))) = (abs‘((𝐹𝑥) − (𝐹𝑤))))
76breq1d 5153 . . . . . . 7 (𝑓 = 𝐹 → ((abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦 ↔ (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))
87imbi2d 340 . . . . . 6 (𝑓 = 𝐹 → (((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))
98rexralbidv 3223 . . . . 5 (𝑓 = 𝐹 → (∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))
1092ralbidv 3221 . . . 4 (𝑓 = 𝐹 → (∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))
1110elrab 3692 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐵m 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)} ↔ (𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))
122, 11bitrdi 287 . 2 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))
13 cnex 11236 . . . . 5 ℂ ∈ V
1413ssex 5321 . . . 4 (𝐵 ⊆ ℂ → 𝐵 ∈ V)
1513ssex 5321 . . . 4 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
16 elmapg 8879 . . . 4 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵))
1714, 15, 16syl2anr 597 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐵m 𝐴) ↔ 𝐹:𝐴𝐵))
1817anbi1d 631 . 2 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → ((𝐹 ∈ (𝐵m 𝐴) ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))
1912, 18bitrd 279 1 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹:𝐴𝐵 ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  {crab 3436  Vcvv 3480  wss 3951   class class class wbr 5143  wf 6557  cfv 6561  (class class class)co 7431  m cmap 8866  cc 11153   < clt 11295  cmin 11492  +crp 13034  abscabs 15273  cnccncf 24902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8868  df-cncf 24904
This theorem is referenced by:  elcncf2  24916  cncff  24919  elcncf1di  24921  rescncf  24923  cncfmet  24935  cncfshift  45889  cncfperiod  45894
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