Theorem txdis 13713

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 13521	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		distop 13521	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ Top)
3		unipw 4217	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
4	3	eqcomi 2181	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
5		unipw 4217	. . . . . . 7 ⊢ ∪ 𝒫 𝐵 = 𝐵
6	5	eqcomi 2181	. . . . . 6 ⊢ 𝐵 = ∪ 𝒫 𝐵
7	4, 6	txuni 13699	. . . . 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 3210	. . . 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 3971	. . 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 3587	. . . . . . . . 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 6165	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (1st ‘𝑦) ∈ 𝐴)
16		snelpwi 4212	. . . . . . . 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 6166	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (2nd ‘𝑦) ∈ 𝐵)
19		snelpwi 4212	. . . . . . . 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 3624	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
22		1st2nd2 6175	. . . . . . . . . 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 3605	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
25	21, 24	eleqtrid 2266	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
26		simprl 529	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ 𝑥)
27	23, 26	eqeltrrd 2255	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ 𝑥)
28	27	snssd 3737	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥)
29		xpeq1 4640	. . . . . . . . . 10 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑧 × 𝑤) = ({(1st ‘𝑦)} × 𝑤))
30	29	eleq2d 2247	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st ‘𝑦)} × 𝑤)))
31	29	sseq1d 3184	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → ((𝑧 × 𝑤) ⊆ 𝑥 ↔ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥))
32	30, 31	anbi12d 473	. . . . . . . 8 ⊢ (𝑧 = {(1st ‘𝑦)} → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ∧ ({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥)))
33		xpeq2 4641	. . . . . . . . . . 11 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = ({(1st ‘𝑦)} × {(2nd ‘𝑦)}))
34		1stexg 6167	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
35	34	elv 2741	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
36		2ndexg 6168	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (2nd ‘𝑦) ∈ V)
37	36	elv 2741	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑦) ∈ V
38	35, 37	xpsn 5692	. . . . . . . . . . 11 ⊢ ({(1st ‘𝑦)} × {(2nd ‘𝑦)}) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}
39	33, 38	eqtrdi 2226	. . . . . . . . . 10 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
40	39	eleq2d 2247	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}))
41	39	sseq1d 3184	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (({(1st ‘𝑦)} × 𝑤) ⊆ 𝑥 ↔ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥))
42	40, 41	anbi12d 473	. . . . . . . 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 2856	. . . . . . 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 1242	. . . . . 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 2557	. . . 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 13695	. . . . 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 3161	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
51	12, 50	eqssd 3172	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  ∀wral 2455  ∃wrex 2456  Vcvv 2737   ⊆ wss 3129  𝒫 cpw 3575  {csn 3592  ⟨cop 3595  ∪ cuni 3809   × cxp 4624  ‘cfv 5216  (class class class)co 5874  1^st c1st 6138  2^nd c2nd 6139  Topctop 13433   ×_t ctx 13688

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-coll 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536

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-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  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-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-topgen 12708  df-top 13434  df-topon 13447  df-bases 13479  df-tx 13689

This theorem is referenced by: (None)