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Theorem elcncf 22911
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 22910 . . . 4 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21eleq2d 2835 . . 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 6331 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
4 fveq1 6331 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑓𝑤) = (𝐹𝑤))
53, 4oveq12d 6810 . . . . . . . . 9 (𝑓 = 𝐹 → ((𝑓𝑥) − (𝑓𝑤)) = ((𝐹𝑥) − (𝐹𝑤)))
65fveq2d 6336 . . . . . . . 8 (𝑓 = 𝐹 → (abs‘((𝑓𝑥) − (𝑓𝑤))) = (abs‘((𝐹𝑥) − (𝐹𝑤))))
76breq1d 4794 . . . . . . 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 3205 . . . . 5 (𝑓 = 𝐹 → (∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝐹𝑥) − (𝐹𝑤))) < 𝑦)))
1092ralbidv 3137 . . . 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 3513 . . 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 10218 . . . . 5 ℂ ∈ V
1413ssex 4933 . . . 4 (𝐵 ⊆ ℂ → 𝐵 ∈ V)
1513ssex 4933 . . . 4 (𝐴 ⊆ ℂ → 𝐴 ∈ V)
16 elmapg 8021 . . . 4 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐹 ∈ (𝐵𝑚 𝐴) ↔ 𝐹:𝐴𝐵))
1714, 15, 16syl2anr 576 . . 3 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐹 ∈ (𝐵𝑚 𝐴) ↔ 𝐹:𝐴𝐵))
1817anbi1d 607 . 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 382   = wceq 1630  wcel 2144  wral 3060  wrex 3061  {crab 3064  Vcvv 3349  wss 3721   class class class wbr 4784  wf 6027  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  cc 10135   < clt 10275  cmin 10467  +crp 12034  abscabs 14181  cnccncf 22898
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-cncf 22900
This theorem is referenced by:  elcncf2  22912  cncff  22915  elcncf1di  22917  rescncf  22919  cncfmet  22930  cncfshift  40599  cncfperiod  40604
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