Theorem txdis 23585

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 22948	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		distop 22948	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ Top)
3		unipw 5391	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
4	3	eqcomi 2744	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
5		unipw 5391	. . . . . . 7 ⊢ ∪ 𝒫 𝐵 = 𝐵
6	5	eqcomi 2744	. . . . . 6 ⊢ 𝐵 = ∪ 𝒫 𝐵
7	4, 6	txuni 23545	. . . . 5 ⊢ ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝐴 × 𝐵) = ∪ (𝒫 𝐴 ×t 𝒫 𝐵))
8	1, 2, 7	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) = ∪ (𝒫 𝐴 ×t 𝒫 𝐵))
9		eqimss2 3976	. . . 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 5031	. . 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 4541	. . . . . . . . 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 7963	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (1st ‘𝑦) ∈ 𝐴)
16		snelpwi 5385	. . . . . . . 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 7964	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (2nd ‘𝑦) ∈ 𝐵)
19		snelpwi 5385	. . . . . . . 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 4597	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
22		1st2nd2 7970	. . . . . . . . . 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 4569	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
25	21, 24	eleqtrid 2841	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
26		simprl 771	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ 𝑥)
27	23, 26	eqeltrrd 2836	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ 𝑥)
28	27	snssd 4720	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥)
29		xpeq1 5634	. . . . . . . . . 10 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑧 × 𝑤) = ({(1st ‘𝑦)} × 𝑤))
30	29	eleq2d 2821	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st ‘𝑦)} × 𝑤)))
31	29	sseq1d 3948	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → ((𝑧 × 𝑤) ⊆ 𝑥 ↔ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥))
32	30, 31	anbi12d 633	. . . . . . . 8 ⊢ (𝑧 = {(1st ‘𝑦)} → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ∧ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥)))
33		xpeq2 5641	. . . . . . . . . . 11 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = ({(1st ‘𝑦)} × {(2nd ‘𝑦)}))
34		fvex 6842	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
35		fvex 6842	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑦) ∈ V
36	34, 35	xpsn 7083	. . . . . . . . . . 11 ⊢ ({(1st ‘𝑦)} × {(2nd ‘𝑦)}) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}
37	33, 36	eqtrdi 2786	. . . . . . . . . 10 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
38	37	eleq2d 2821	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}))
39	37	sseq1d 3948	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥 ↔ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥))
40	38, 39	anbi12d 633	. . . . . . . 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 3575	. . . . . . 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 1377	. . . . . 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 3135	. . . 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 23521	. . . . 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 597	. . . 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 3923	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
49	12, 48	eqssd 3934	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3049  ∃wrex 3059   ⊆ wss 3885  𝒫 cpw 4531  {csn 4557  ⟨cop 4563  ∪ cuni 4840   × cxp 5618  ‘cfv 6487  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  Topctop 22846   ×_t ctx 23513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2184  ax-ext 2707  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2931  df-ral 3050  df-rex 3060  df-reu 3341  df-rab 3388  df-v 3429  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6443  df-fun 6489  df-fn 6490  df-f 6491  df-f1 6492  df-fo 6493  df-f1o 6494  df-fv 6495  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-topgen 17395  df-top 22847  df-topon 22864  df-bases 22899  df-tx 23515

This theorem is referenced by:  distgp  24052