Theorem cncfval 14029

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 7934	. . 3 ⊢ ℂ ∈ V
2	1	elpw2 4157	. 2 ⊢ (𝐴 ∈ 𝒫 ℂ ↔ 𝐴 ⊆ ℂ)
3	1	elpw2 4157	. 2 ⊢ (𝐵 ∈ 𝒫 ℂ ↔ 𝐵 ⊆ ℂ)
4		mapvalg 6657	. . . . . 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 6653	. . . . 5 ⊢ ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → {𝑓 ∣ 𝑓:𝐴⟶𝐵} ∈ V)
7	5, 6	eqeltrd 2254	. . . 4 ⊢ ((𝐴 ∈ 𝒫 ℂ ∧ 𝐵 ∈ 𝒫 ℂ) → (𝐵 ↑𝑚 𝐴) ∈ V)
8		rabexg 4146	. . . 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 5882	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝑏 ↑𝑚 𝑎) = (𝑏 ↑𝑚 𝐴))
11		raleq 2672	. . . . . . . 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 2478	. . . . . . 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 2477	. . . . . 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 2680	. . . . 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 2732	. . . 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 5881	. . . . 5 ⊢ (𝑏 = 𝐵 → (𝑏 ↑𝑚 𝐴) = (𝐵 ↑𝑚 𝐴))
17	16	rabeqdv 2731	. . . 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 14028	. . . 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 6008	. . 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 1338	. 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)})

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1353   ∈ wcel 2148  {cab 2163  ∀wral 2455  ∃wrex 2456  {crab 2459  Vcvv 2737   ⊆ wss 3129  𝒫 cpw 3575   class class class wbr 4003  ⟶wf 5212  ‘cfv 5216  (class class class)co 5874   ↑_𝑚 cmap 6647  ℂcc 7808   < clt 7991   − cmin 8127  ℝ⁺crp 9652  abscabs 11005  –cn→ccncf 14027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-map 6649  df-cncf 14028

This theorem is referenced by:  elcncf  14030