Theorem txdis 15142

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 14950	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ Top)
2		distop 14950	. . . . 5 ⊢ (𝐵 ∈ 𝑊 → 𝒫 𝐵 ∈ Top)
3		unipw 4333	. . . . . . 7 ⊢ ∪ 𝒫 𝐴 = 𝐴
4	3	eqcomi 2236	. . . . . 6 ⊢ 𝐴 = ∪ 𝒫 𝐴
5		unipw 4333	. . . . . . 7 ⊢ ∪ 𝒫 𝐵 = 𝐵
6	5	eqcomi 2236	. . . . . 6 ⊢ 𝐵 = ∪ 𝒫 𝐵
7	4, 6	txuni 15128	. . . . 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 3293	. . . 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 4076	. . 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 3681	. . . . . . . . 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 6359	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (1st ‘𝑦) ∈ 𝐴)
16		snelpwi 4327	. . . . . . . 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 6360	. . . . . . . 8 ⊢ (𝑦 ∈ (𝐴 × 𝐵) → (2nd ‘𝑦) ∈ 𝐵)
19		snelpwi 4327	. . . . . . . 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 3721	. . . . . . . 8 ⊢ 𝑦 ∈ {𝑦}
22		1st2nd2 6369	. . . . . . . . . 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 3702	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
25	21, 24	eleqtrid 2321	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
26		simprl 531	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ 𝑥)
27	23, 26	eqeltrrd 2310	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ ∈ 𝑥)
28	27	snssd 3839	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ (𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩} ⊆ 𝑥)
29		xpeq1 4763	. . . . . . . . . 10 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑧 × 𝑤) = ({(1st ‘𝑦)} × 𝑤))
30	29	eleq2d 2302	. . . . . . . . 9 ⊢ (𝑧 = {(1st ‘𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st ‘𝑦)} × 𝑤)))
31	29	sseq1d 3267	. . . . . . . . 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 4764	. . . . . . . . . . 11 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = ({(1st ‘𝑦)} × {(2nd ‘𝑦)}))
34		1stexg 6361	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
35	34	elv 2817	. . . . . . . . . . . 12 ⊢ (1st ‘𝑦) ∈ V
36		2ndexg 6362	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ V → (2nd ‘𝑦) ∈ V)
37	36	elv 2817	. . . . . . . . . . . 12 ⊢ (2nd ‘𝑦) ∈ V
38	35, 37	xpsn 5854	. . . . . . . . . . 11 ⊢ ({(1st ‘𝑦)} × {(2nd ‘𝑦)}) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}
39	33, 38	eqtrdi 2281	. . . . . . . . . 10 ⊢ (𝑤 = {(2nd ‘𝑦)} → ({(1st ‘𝑦)} × 𝑤) = {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩})
40	39	eleq2d 2302	. . . . . . . . 9 ⊢ (𝑤 = {(2nd ‘𝑦)} → (𝑦 ∈ ({(1st ‘𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st ‘𝑦), (2nd ‘𝑦)⟩}))
41	39	sseq1d 3267	. . . . . . . . 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 2936	. . . . . . 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 1278	. . . . . 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 2622	. . . 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 15124	. . . . 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 3244	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
51	12, 50	eqssd 3255	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  Vcvv 2813   ⊆ wss 3211  𝒫 cpw 3669  {csn 3689  ⟨cop 3692  ∪ cuni 3914   × cxp 4747  ‘cfv 5352  (class class class)co 6050  1^st c1st 6332  2^nd c2nd 6333  Topctop 14862   ×_t ctx 15117

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-topgen 13473  df-top 14863  df-topon 14876  df-bases 14908  df-tx 15118

This theorem is referenced by: (None)