Theorem txdis 15378

Description: The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
txdis	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))

Proof of Theorem txdis

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		distop 15186	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		distop 15186	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ Top)
3		unipw 4357	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
4	3	eqcomi 2242	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
5		unipw 4357	. . . . . . 7 ⊢ ∪ 𝒫 𝐵 = 𝐵
6	5	eqcomi 2242	. . . . . 6 ⊢ 𝐵 = ∪ 𝒫 𝐵
7	4, 6	txuni 15364	. . . . 5 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝐴 × 𝐵) = ∪ (𝒫 𝐴 ×t 𝒫 𝐵))
8	1, 2, 7	syl2an 289	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) = ∪ (𝒫 𝐴 ×t 𝒫 𝐵))
9		eqimss2 3303	. . . 4 ⊢ ((𝐴 × 𝐵) = ∪ (𝒫 𝐴 ×t 𝒫 𝐵) → ∪ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
10	8, 9	syl 14	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
11		sspwuni 4097	. . 3 ⊢ ((𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵) ↔ ∪ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
12	10, 11	sylibr 134	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵))
13		elelpwi 3701	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵)) → 𝑦 ∈ (𝐴 × 𝐵))
14	13	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ (𝐴 × 𝐵))
15		xp1st 6399	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (1st ‘𝑦) ∈ 𝐴)
16		snelpwi 4351	. . . . . . . 8 ⊢ ((1st ‘𝑦) ∈ 𝐴 → {(1st ‘𝑦)} ∈ 𝒫 𝐴)
17	14, 15, 16	3syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(1st ‘𝑦)} ∈ 𝒫 𝐴)
18		xp2nd 6400	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (2nd ‘𝑦) ∈ 𝐵)
19		snelpwi 4351	. . . . . . . 8 ⊢ ((2nd ‘𝑦) ∈ 𝐵 → {(2nd ‘𝑦)} ∈ 𝒫 𝐵)
20	14, 18, 19	3syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(2nd ‘𝑦)} ∈ 𝒫 𝐵)
21		vsnid 3741	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
22		1st2nd2 6409	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
23	14, 22	syl 14	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
24	23	sneqd 3722	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
25	21, 24	eleqtrid 2327	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
26		simprl 535	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ 𝑥)
27	23, 26	eqeltrrd 2316	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ 𝑥)
28	27	snssd 3860	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥)
29		xpeq1 4788	. . . . . . . . . 10 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑧 × 𝑤) = ({(1st ‘𝑦)} × 𝑤))
30	29	eleq2d 2308	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st ‘𝑦)} × 𝑤)))
31	29	sseq1d 3277	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → ((𝑧 × 𝑤) ⊆ 𝑥 ↔ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥))
32	30, 31	anbi12d 477	. . . . . . . 8 ⊢ (𝑧 = {(1st ‘𝑦)} → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ∧ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥)))
33		xpeq2 4789	. . . . . . . . . . 11 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = ({(1st ‘𝑦)} × {(2nd ‘𝑦)}))
34		1stexg 6401	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
35	34	elv 2825	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
36		2ndexg 6402	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (2nd ‘𝑦) ∈ V)
37	36	elv 2825	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑦) ∈ V
38	35, 37	xpsn 5885	. . . . . . . . . . 11 ⊢ ({(1st ‘𝑦)} × {(2nd ‘𝑦)}) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}
39	33, 38	eqtrdi 2287	. . . . . . . . . 10 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
40	39	eleq2d 2308	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}))
41	39	sseq1d 3277	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥 ↔ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥))
42	40, 41	anbi12d 477	. . . . . . . 8 ⊢ (𝑤 = {(2nd ‘𝑦)} → ((𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ∧ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ∧ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥)))
43	32, 42	rspc2ev 2945	. . . . . . 7 ⊢ (({(1st ‘𝑦)} ∈ 𝒫 𝐴 ∧ {(2nd ‘𝑦)} ∈ 𝒫 𝐵 ∧ (𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ∧ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥)) → ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
44	17, 20, 25, 28, 43	syl112anc 1282	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
45	44	expr 375	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
46	45	ralrimdva 2630	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
47		eltx 15360	. . . . 5 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
48	1, 2, 47	syl2an 289	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵) ↔ ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
49	46, 48	sylibrd 169	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → 𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵)))
50	49	ssrdv 3254	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
51	12, 50	eqssd 3265	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688  {csn 3709  ⟨cop 3712  ∪ cuni 3935   × cxp 4772  ‘cfv 5377  (class class class)co 6085  1^st c1st 6372  2^nd c2nd 6373  Topctop 15098   ×_t ctx 15353

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-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

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-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  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-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-topgen 13614  df-top 15099  df-topon 15112  df-bases 15144  df-tx 15354

This theorem is used by: (None)