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Theorem elcncf 22739
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 22738 . . . 4 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21eleq2d 2716 . . 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 6228 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
4 fveq1 6228 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑤) = (𝐹𝑤))
53, 4oveq12d 6708 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) − (𝑓𝑤)) = ((𝐹𝑥) − (𝐹𝑤)))
65fveq2d 6233 . . . . . . . 8 (𝑓 = 𝐹 → (abs‘((𝑓𝑥) − (𝑓𝑤))) = (abs‘((𝐹𝑥) − (𝐹𝑤))))
76breq1d 4695 . . . . . . 7 (𝑓 = 𝐹 → ((abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦 ↔ (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))
87imbi2d 329 . . . . . 6 (𝑓 = 𝐹 → (((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))
98rexralbidv 3087 . . . . 5 (𝑓 = 𝐹 → (∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))
1092ralbidv 3018 . . . 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 3396 . . 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, 11syl6bb 276 . 2 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐴cn𝐵) ↔ (𝐹 ∈ (𝐵𝑚 𝐴) ∧ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦))))
13 cnex 10055 . . . . 5 ℂ ∈ V
1413ssex 4835 . . . 4 (𝐵 ⊆ ℂ → 𝐵 ∈ V)
1513ssex 4835 . . . 4 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
16 elmapg 7912 . . . 4 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐹 ∈ (𝐵𝑚 𝐴) ↔ 𝐹:𝐴𝐵))
1714, 15, 16syl2anr 494 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐵𝑚 𝐴) ↔ 𝐹:𝐴𝐵))
1817anbi1d 741 . 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 268 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 196  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607   class class class wbr 4685  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  cc 9972   < clt 10112  cmin 10304  +crp 11870  abscabs 14018  cnccncf 22726
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-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  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-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-cncf 22728
This theorem is referenced by:  elcncf2  22740  cncff  22743  elcncf1di  22745  rescncf  22747  cncfmet  22758  cncfshift  40405  cncfperiod  40410
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