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Theorem cncfval 12737
Description: The value of the continuous complex function operation is the set of continuous functions from 𝐴 to 𝐵. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
Assertion
Ref Expression
cncfval ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
Distinct variable groups:   𝑤,𝑓,𝑥,𝑦,𝑧,𝐴   𝐵,𝑓,𝑤,𝑥,𝑦,𝑧

Proof of Theorem cncfval
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnex 7751 . . 3 ℂ ∈ V
21elpw2 4082 . 2 (𝐴 ∈ 𝒫 ℂ ↔ 𝐴 ⊆ ℂ)
31elpw2 4082 . 2 (𝐵 ∈ 𝒫 ℂ ↔ 𝐵 ⊆ ℂ)
4 mapvalg 6552 . . . . . 6 ((𝐵 ∈ 𝒫 ℂ ∧ 𝐴 ∈ 𝒫 ℂ) → (𝐵𝑚 𝐴) = {𝑓𝑓:𝐴𝐵})
54ancoms 266 . . . . 5 ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → (𝐵𝑚 𝐴) = {𝑓𝑓:𝐴𝐵})
6 mapex 6548 . . . . 5 ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → {𝑓𝑓:𝐴𝐵} ∈ V)
75, 6eqeltrd 2216 . . . 4 ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → (𝐵𝑚 𝐴) ∈ V)
8 rabexg 4071 . . . 4 ((𝐵𝑚 𝐴) ∈ V → {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)} ∈ V)
97, 8syl 14 . . 3 ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)} ∈ V)
10 oveq2 5782 . . . . 5 (𝑎 = 𝐴 → (𝑏𝑚 𝑎) = (𝑏𝑚 𝐴))
11 raleq 2626 . . . . . . . 8 (𝑎 = 𝐴 → (∀𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ∀𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)))
1211rexbidv 2438 . . . . . . 7 (𝑎 = 𝐴 → (∃𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)))
1312ralbidv 2437 . . . . . 6 (𝑎 = 𝐴 → (∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)))
1413raleqbi1dv 2634 . . . . 5 (𝑎 = 𝐴 → (∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦) ↔ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)))
1510, 14rabeqbidv 2681 . . . 4 (𝑎 = 𝐴 → {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)} = {𝑓 ∈ (𝑏𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
16 oveq1 5781 . . . . 5 (𝑏 = 𝐵 → (𝑏𝑚 𝐴) = (𝐵𝑚 𝐴))
1716rabeqdv 2680 . . . 4 (𝑏 = 𝐵 → {𝑓 ∈ (𝑏𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)} = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
18 df-cncf 12736 . . . 4 cn→ = (𝑎 ∈ 𝒫 ℂ, 𝑏 ∈ 𝒫 ℂ ↦ {𝑓 ∈ (𝑏𝑚 𝑎) ∣ ∀𝑥𝑎𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑎 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
1915, 17, 18ovmpog 5905 . . 3 ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ ∧ {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)} ∈ V) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
209, 19mpd3an3 1316 . 2 ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
212, 3, 20syl2anbr 290 1 ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴cn𝐵) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ∀𝑥𝐴𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝐴 ((abs‘(𝑥𝑤)) < 𝑧 → (abs‘((𝑓𝑥) − (𝑓𝑤))) < 𝑦)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1331  wcel 1480  {cab 2125  wral 2416  wrex 2417  {crab 2420  Vcvv 2686  wss 3071  𝒫 cpw 3510   class class class wbr 3929  wf 5119  cfv 5123  (class class class)co 5774  𝑚 cmap 6542  cc 7625   < clt 7807  cmin 7940  +crp 9448  abscabs 10776  cnccncf 12735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-cncf 12736
This theorem is referenced by:  elcncf  12738
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