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Theorem cncfrss 25011
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 24998 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏m 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpocl1 7642 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ∈ 𝒫 ℂ)
32elpwid 4567 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  wral 3079  wrex 3089  {crab 3417  wss 3907  𝒫 cpw 4558   class class class wbr 5105  cfv 6525  (class class class)co 7400  m cmap 8812  cc 11086   < clt 11231  cmin 11429  +crp 13007  abscabs 15275  cnccncf 24996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-dm 5662  df-iota 6481  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-cncf 24998
This theorem is referenced by:  cncff  25013  cncfi  25014  rescncf  25017  cncfcdm  25018  cncfco  25027  cncfcompt2  25028  cncfmpt2f  25035  cncfcnvcn  25045  mulcncf  25566  cncombf  25778  cnlimci  26009  ulmcn  26520  efmul2picn  34900  mulcncff  46442  subcncff  46452  negcncfg  46453  addcncff  46456  ioccncflimc  46457  icocncflimc  46461  divcncff  46463
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