Theorem cncfrss2 25012

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 24998	. . 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 7643	. 2 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ∈ 𝒫 ℂ)
3	2	elpwid 4567	1 ⊢ (𝐹 ∈ (𝐴–cn→𝐵) → 𝐵 ⊆ ℂ)

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  ℂcc 11086   < clt 11231   − cmin 11429  ℝ⁺crp 13007  abscabs 15275  –cn→ccncf 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  climcncf  25020  cncfco  25027  cncfcnvcn  25045  cnlimci  26009  cncfmptssg  46443  cncfcompt  46455