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Theorem cncfrss2 24407
Description: Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
Assertion
Ref Expression
cncfrss2 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)

Proof of Theorem cncfrss2
Dummy variables 𝑎 𝑏 𝑓 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cncf 24393 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏m 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpocl2 7649 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ∈ 𝒫 ℂ)
32elpwid 4611 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  wral 3061  wrex 3070  {crab 3432  wss 3948  𝒫 cpw 4602   class class class wbr 5148  cfv 6543  (class class class)co 7408  m cmap 8819  cc 11107   < clt 11247  cmin 11443  +crp 12973  abscabs 15180  cnccncf 24391
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-iota 6495  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-cncf 24393
This theorem is referenced by:  cncff  24408  cncfi  24409  rescncf  24412  climcncf  24415  cncfco  24422  cncfcnvcn  24440  cnlimci  25405  cncfmptssg  44577  cncfcompt  44589
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