Theorem cncfval 15675

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) → (𝐴–cn→𝐵) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 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.

Step	Hyp	Ref	Expression
1		cnex 8303	. . 3 ⊢ ℂ ∈ V
2	1	elpw2 4293	. 2 ⊢ (𝐴 ∈ 𝒫 ℂ ↔ 𝐴 ⊆ ℂ)
3	1	elpw2 4293	. 2 ⊢ (𝐵 ∈ 𝒫 ℂ ↔ 𝐵 ⊆ ℂ)
4		mapvalg 6932	. . . . . 6 ⊢ ((𝐵 ∈ 𝒫 ℂ ∧ 𝐴 ∈ 𝒫 ℂ) → (𝐵 ↑𝑚 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶𝐵})
5	4	ancoms 268	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → (𝐵 ↑𝑚 𝐴) = {𝑓 ∣ 𝑓:𝐴⟶𝐵})
6		mapex 6928	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
7	5, 6	eqeltrd 2315	. . . 4 ⊢ ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → (𝐵 ↑𝑚 𝐴) ∈ V)
8		rabexg 4279	. . . 4 ⊢ ((𝐵 ↑𝑚 𝐴) ∈ V → {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)} ∈ V)
9	7, 8	syl 14	. . 3 ⊢ ((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)} ∈ V)
10		oveq2 6093	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑏 ↑𝑚 𝑎) = (𝑏 ↑𝑚 𝐴))
11		raleq 2749	. . . . . . . 8 ⊢ (𝑎 = 𝐴 → (∀𝑤 ∈ 𝑎 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦) ↔ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)))
12	11	rexbidv 2551	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑎 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦) ↔ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)))
13	12	ralbidv 2550	. . . . . 6 ⊢ (𝑎 = 𝐴 → (∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑎 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦) ↔ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)))
14	13	raleqbi1dv 2761	. . . . 5 ⊢ (𝑎 = 𝐴 → (∀𝑥 ∈ 𝑎 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑎 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)))
15	10, 14	rabeqbidv 2816	. . . 4 ⊢ (𝑎 = 𝐴 → {𝑓 ∈ (𝑏 ↑𝑚 𝑎) ∣ ∀𝑥 ∈ 𝑎 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑎 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)} = {𝑓 ∈ (𝑏 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)})
16		oveq1 6092	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 ↑𝑚 𝐴) = (𝐵 ↑𝑚 𝐴))
17	16	rabeqdv 2815	. . . 4 ⊢ (𝑏 = 𝐵 → {𝑓 ∈ (𝑏 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)} = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)})
18		df-cncf 15674	. . . 4 ⊢ –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)})
19	15, 17, 18	ovmpog 6223	. . 3 ⊢ ((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)} ∈ V) → (𝐴–cn→𝐵) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)})
20	9, 19	mpd3an3 1379	. 2 ⊢ ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → (𝐴–cn→𝐵) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)})
21	2, 3, 20	syl2anbr 292	1 ⊢ ((𝐴 ⊆ ℂ ∧ 𝐵 ⊆ ℂ) → (𝐴–cn→𝐵) = {𝑓 ∈ (𝐵 ↑𝑚 𝐴) ∣ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝐴 ((abs‘(𝑥 − 𝑤)) < 𝑧 → (abs‘((𝑓‘𝑥) − (𝑓‘𝑤))) < 𝑦)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  {cab 2224  ∀wral 2528  ∃wrex 2529  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688   class class class wbr 4130  ⟶wf 5373  ‘cfv 5377  (class class class)co 6085   ↑_𝑚 cmap 6922  ℂcc 8177   < clt 8360   − cmin 8497  ℝ⁺crp 10056  abscabs 11765  –cn→ccncf 15673

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-map 6924  df-cncf 15674

This theorem is used by:  elcncf  15676