Theorem cncfrss2 24934

Description: Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)

Assertion

Ref	Expression
cncfrss2	⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)

Proof of Theorem cncfrss2

Dummy variables 𝑎 𝑏 𝑓 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-cncf 24920	. . 3 ⊢ –cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏 ↑m 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑎 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)})
2	1	elmpocl2 7635	. 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ∈ 𝒫 ℂ)
3	2	elpwid 4563	1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  {crab 3413   ⊆ wss 3904  𝒫 cpw 4554   class class class wbr 5099  ‘cfv 6517  (class class class)co 7392   ↑_m cmap 8803  ℂcc 11068   < clt 11213   − cmin 11411  ℝ⁺crp 12990  abscabs 15244  –cn→ccncf 24918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-xp 5651  df-dm 5655  df-iota 6473  df-fv 6525  df-ov 7395  df-oprab 7396  df-mpo 7397  df-cncf 24920

This theorem is referenced by:  cncff  24935  cncfi  24936  rescncf  24939  climcncf  24942  cncfco  24949  cncfcnvcn  24967  cnlimci  25931  cncfmptssg  46409  cncfcompt  46421