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Theorem cncfrss 23491
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 23478 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏m 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpocl1 7380 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ∈ 𝒫 ℂ)
32elpwid 4551 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐴 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  wral 3136  wrex 3137  {crab 3140  wss 3934  𝒫 cpw 4537   class class class wbr 5057  cfv 6348  (class class class)co 7148  m cmap 8398  cc 10527   < clt 10667  cmin 10862  +crp 12381  abscabs 14585  cnccncf 23476
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558  df-iota 6307  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-cncf 23478
This theorem is referenced by:  cncff  23493  cncfi  23494  rescncf  23497  cncffvrn  23498  cncfco  23507  cncfmpt2f  23514  cncfcnvcn  23521  cncombf  24251  cnlimci  24479  ulmcn  24979  efmul2picn  31860  mulcncff  42140  subcncff  42152  negcncfg  42153  addcncff  42156  ioccncflimc  42157  icocncflimc  42161  divcncff  42163  cncfcompt2  42171
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