MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cncfrss Structured version   Visualization version   GIF version

Theorem cncfrss 23064
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 23051 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpt2cl1 7137 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ∈ 𝒫 ℂ)
32elpwid 4390 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2164  wral 3117  wrex 3118  {crab 3121  wss 3798  𝒫 cpw 4378   class class class wbr 4873  cfv 6123  (class class class)co 6905  𝑚 cmap 8122  cc 10250   < clt 10391  cmin 10585  +crp 12112  abscabs 14351  cnccncf 23049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-xp 5348  df-dm 5352  df-iota 6086  df-fv 6131  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-cncf 23051
This theorem is referenced by:  cncff  23066  cncfi  23067  rescncf  23070  cncffvrn  23071  cncfco  23080  cncfmpt2f  23087  cncfcnvcn  23094  cncombf  23824  cnlimci  24052  ulmcn  24552  efmul2picn  31212  mulcncff  40869  subcncff  40881  negcncfg  40882  addcncff  40885  ioccncflimc  40886  icocncflimc  40890  divcncff  40892  cncfcompt2  40900
  Copyright terms: Public domain W3C validator