MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cncfrss2 Structured version   Visualization version   GIF version

Theorem cncfrss2 24834
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 24820 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏m 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpocl2 7648 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ∈ 𝒫 ℂ)
32elpwid 4584 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wral 3051  wrex 3060  {crab 3415  wss 3926  𝒫 cpw 4575   class class class wbr 5119  cfv 6530  (class class class)co 7403  m cmap 8838  cc 11125   < clt 11267  cmin 11464  +crp 13006  abscabs 15251  cnccncf 24818
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-xp 5660  df-dm 5664  df-iota 6483  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-cncf 24820
This theorem is referenced by:  cncff  24835  cncfi  24836  rescncf  24839  climcncf  24842  cncfco  24849  cncfcnvcn  24868  cnlimci  25840  cncfmptssg  45848  cncfcompt  45860
  Copyright terms: Public domain W3C validator