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Theorem cncfrss2 24632
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 24618 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏m 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpocl2 7652 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ∈ 𝒫 ℂ)
32elpwid 4610 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2104  wral 3059  wrex 3068  {crab 3430  wss 3947  𝒫 cpw 4601   class class class wbr 5147  cfv 6542  (class class class)co 7411  m cmap 8822  cc 11110   < clt 11252  cmin 11448  +crp 12978  abscabs 15185  cnccncf 24616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-xp 5681  df-dm 5685  df-iota 6494  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-cncf 24618
This theorem is referenced by:  cncff  24633  cncfi  24634  rescncf  24637  climcncf  24640  cncfco  24647  cncfcnvcn  24666  cnlimci  25638  cncfmptssg  44885  cncfcompt  44897
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