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Theorem cncfrss2 23066
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 23052 . . 3 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
21elmpt2cl2 7139 . 2 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ∈ 𝒫 ℂ)
32elpwid 4391 1 (𝐹 ∈ (𝐴cn𝐵) → 𝐵 ⊆ ℂ)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2166  wral 3118  wrex 3119  {crab 3122  wss 3799  𝒫 cpw 4379   class class class wbr 4874  cfv 6124  (class class class)co 6906  𝑚 cmap 8123  cc 10251   < clt 10392  cmin 10586  +crp 12113  abscabs 14352  cnccncf 23050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ral 3123  df-rex 3124  df-rab 3127  df-v 3417  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4660  df-br 4875  df-opab 4937  df-xp 5349  df-dm 5353  df-iota 6087  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-cncf 23052
This theorem is referenced by:  cncff  23067  cncfi  23068  rescncf  23071  climcncf  23074  cncfco  23081  cncfcnvcn  23095  cnlimci  24053  cncfmptssg  40879  cncfcompt  40892
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