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Theorem cncfrss 24931
Description: Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
Assertion
Ref Expression
cncfrss (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)

Proof of Theorem cncfrss
Dummy variables 𝑎 𝑏 𝑓 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cncf 24918 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏m 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpocl1 7675 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ∈ 𝒫 ℂ)
32elpwid 4614 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wral 3059  wrex 3068  {crab 3433  wss 3963  𝒫 cpw 4605   class class class wbr 5148  cfv 6563  (class class class)co 7431  m cmap 8865  cc 11151   < clt 11293  cmin 11490  +crp 13032  abscabs 15270  cnccncf 24916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-iota 6516  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-cncf 24918
This theorem is referenced by:  cncff  24933  cncfi  24934  rescncf  24937  cncfcdm  24938  cncfco  24947  cncfcompt2  24948  cncfmpt2f  24955  cncfcnvcn  24966  mulcncf  25494  cncombf  25707  cnlimci  25939  ulmcn  26457  efmul2picn  34590  mulcncff  45826  subcncff  45836  negcncfg  45837  addcncff  45840  ioccncflimc  45841  icocncflimc  45845  divcncff  45847
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