Theorem txdis 23553

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 22916	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		distop 22916	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ Top)
3		unipw 5405	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
4	3	eqcomi 2738	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
5		unipw 5405	. . . . . . 7 ⊢ ∪ 𝒫 𝐵 = 𝐵
6	5	eqcomi 2738	. . . . . 6 ⊢ 𝐵 = ∪ 𝒫 𝐵
7	4, 6	txuni 23513	. . . . 5 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝐴 × 𝐵) = ∪ (𝒫 𝐴 ×t 𝒫 𝐵))
8	1, 2, 7	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) = ∪ (𝒫 𝐴 ×t 𝒫 𝐵))
9		eqimss2 4003	. . . 4 ⊢ ((𝐴 × 𝐵) = ∪ (𝒫 𝐴 ×t 𝒫 𝐵) → ∪ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
10	8, 9	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∪ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
11		sspwuni 5059	. . 3 ⊢ ((𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵) ↔ ∪ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
12	10, 11	sylibr 234	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵))
13		elelpwi 4569	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵)) → 𝑦 ∈ (𝐴 × 𝐵))
14	13	adantl 481	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ (𝐴 × 𝐵))
15		xp1st 7979	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (1st ‘𝑦) ∈ 𝐴)
16		snelpwi 5398	. . . . . . . 8 ⊢ ((1st ‘𝑦) ∈ 𝐴 → {(1st ‘𝑦)} ∈ 𝒫 𝐴)
17	14, 15, 16	3syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(1st ‘𝑦)} ∈ 𝒫 𝐴)
18		xp2nd 7980	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (2nd ‘𝑦) ∈ 𝐵)
19		snelpwi 5398	. . . . . . . 8 ⊢ ((2nd ‘𝑦) ∈ 𝐵 → {(2nd ‘𝑦)} ∈ 𝒫 𝐵)
20	14, 18, 19	3syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(2nd ‘𝑦)} ∈ 𝒫 𝐵)
21		vsnid 4623	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
22		1st2nd2 7986	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
23	14, 22	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
24	23	sneqd 4597	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
25	21, 24	eleqtrid 2834	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
26		simprl 770	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ 𝑥)
27	23, 26	eqeltrrd 2829	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ 𝑥)
28	27	snssd 4769	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥)
29		xpeq1 5645	. . . . . . . . . 10 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑧 × 𝑤) = ({(1st ‘𝑦)} × 𝑤))
30	29	eleq2d 2814	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st ‘𝑦)} × 𝑤)))
31	29	sseq1d 3975	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → ((𝑧 × 𝑤) ⊆ 𝑥 ↔ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥))
32	30, 31	anbi12d 632	. . . . . . . 8 ⊢ (𝑧 = {(1st ‘𝑦)} → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ∧ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥)))
33		xpeq2 5652	. . . . . . . . . . 11 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = ({(1st ‘𝑦)} × {(2nd ‘𝑦)}))
34		fvex 6853	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
35		fvex 6853	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑦) ∈ V
36	34, 35	xpsn 7095	. . . . . . . . . . 11 ⊢ ({(1st ‘𝑦)} × {(2nd ‘𝑦)}) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}
37	33, 36	eqtrdi 2780	. . . . . . . . . 10 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
38	37	eleq2d 2814	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}))
39	37	sseq1d 3975	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥 ↔ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥))
40	38, 39	anbi12d 632	. . . . . . . 8 ⊢ (𝑤 = {(2nd ‘𝑦)} → ((𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ∧ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ∧ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥)))
41	32, 40	rspc2ev 3598	. . . . . . 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))
42	17, 20, 25, 28, 41	syl112anc 1376	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
43	42	expr 456	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝑦 ∈ 𝑥) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
44	43	ralrimdva 3133	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∀𝑦 ∈ 𝑥 ∃𝑧 ∈ 𝒫 𝐴∃𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
45		eltx 23489	. . . . 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)))
46	1, 2, 45	syl2an 596	. . . 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)))
47	44, 46	sylibrd 259	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → 𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵)))
48	47	ssrdv 3949	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
49	12, 48	eqssd 3961	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911  𝒫 cpw 4559  {csn 4585  ⟨cop 4591  ∪ cuni 4867   × cxp 5629  ‘cfv 6499  (class class class)co 7369  1^st c1st 7945  2^nd c2nd 7946  Topctop 22814   ×_t ctx 23481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-topgen 17383  df-top 22815  df-topon 22832  df-bases 22867  df-tx 23483

This theorem is referenced by:  distgp  24020