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Theorem cncfrss 24855
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 24842 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏m 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpocl1 7663 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ∈ 𝒫 ℂ)
32elpwid 4613 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098  wral 3050  wrex 3059  {crab 3418  wss 3944  𝒫 cpw 4604   class class class wbr 5149  cfv 6549  (class class class)co 7419  m cmap 8845  cc 11138   < clt 11280  cmin 11476  +crp 13009  abscabs 15217  cnccncf 24840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-xp 5684  df-dm 5688  df-iota 6501  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-cncf 24842
This theorem is referenced by:  cncff  24857  cncfi  24858  rescncf  24861  cncfcdm  24862  cncfco  24871  cncfcompt2  24872  cncfmpt2f  24879  cncfcnvcn  24890  mulcncf  25418  cncombf  25631  cnlimci  25862  ulmcn  26380  efmul2picn  34359  mulcncff  45396  subcncff  45406  negcncfg  45407  addcncff  45410  ioccncflimc  45411  icocncflimc  45415  divcncff  45417
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