Theorem cncfrss2 24854

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 24840	. . 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 7658	. 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ∈ 𝒫 ℂ)
3	2	elpwid 4589	1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)

Syntax hints:   → wi 4   ∈ wcel 2107  ∀wral 3050  ∃wrex 3059  {crab 3419   ⊆ wss 3931  𝒫 cpw 4580   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413   ↑_m cmap 8848  ℂcc 11135   < clt 11277   − cmin 11474  ℝ⁺crp 13016  abscabs 15255  –cn→ccncf 24838

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-xp 5671  df-dm 5675  df-iota 6494  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpo 7418  df-cncf 24840

This theorem is referenced by:  cncff  24855  cncfi  24856  rescncf  24859  climcncf  24862  cncfco  24869  cncfcnvcn  24888  cnlimci  25860  cncfmptssg  45843  cncfcompt  45855