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Theorem txdis 22983
Description: The topological product of discrete spaces is discrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txdis ((𝐴𝑉𝐵𝑊) → (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.
StepHypRef Expression
1 distop 22345 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 distop 22345 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ∈ Top)
3 unipw 5407 . . . . . . 7 𝒫 𝐴 = 𝐴
43eqcomi 2745 . . . . . 6 𝐴 = 𝒫 𝐴
5 unipw 5407 . . . . . . 7 𝒫 𝐵 = 𝐵
65eqcomi 2745 . . . . . 6 𝐵 = 𝒫 𝐵
74, 6txuni 22943 . . . . 5 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝐴 × 𝐵) = (𝒫 𝐴 ×t 𝒫 𝐵))
81, 2, 7syl2an 596 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) = (𝒫 𝐴 ×t 𝒫 𝐵))
9 eqimss2 4001 . . . 4 ((𝐴 × 𝐵) = (𝒫 𝐴 ×t 𝒫 𝐵) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
108, 9syl 17 . . 3 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
11 sspwuni 5060 . . 3 ((𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵) ↔ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
1210, 11sylibr 233 . 2 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵))
13 elelpwi 4570 . . . . . . . . 9 ((𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵)) → 𝑦 ∈ (𝐴 × 𝐵))
1413adantl 482 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ (𝐴 × 𝐵))
15 xp1st 7953 . . . . . . . 8 (𝑦 ∈ (𝐴 × 𝐵) → (1st𝑦) ∈ 𝐴)
16 snelpwi 5400 . . . . . . . 8 ((1st𝑦) ∈ 𝐴 → {(1st𝑦)} ∈ 𝒫 𝐴)
1714, 15, 163syl 18 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(1st𝑦)} ∈ 𝒫 𝐴)
18 xp2nd 7954 . . . . . . . 8 (𝑦 ∈ (𝐴 × 𝐵) → (2nd𝑦) ∈ 𝐵)
19 snelpwi 5400 . . . . . . . 8 ((2nd𝑦) ∈ 𝐵 → {(2nd𝑦)} ∈ 𝒫 𝐵)
2014, 18, 193syl 18 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(2nd𝑦)} ∈ 𝒫 𝐵)
21 vsnid 4623 . . . . . . . 8 𝑦 ∈ {𝑦}
22 1st2nd2 7960 . . . . . . . . . 10 (𝑦 ∈ (𝐴 × 𝐵) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2314, 22syl 17 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2423sneqd 4598 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st𝑦), (2nd𝑦)⟩})
2521, 24eleqtrid 2844 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩})
26 simprl 769 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦𝑥)
2723, 26eqeltrrd 2839 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝑥)
2827snssd 4769 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)
29 xpeq1 5647 . . . . . . . . . 10 (𝑧 = {(1st𝑦)} → (𝑧 × 𝑤) = ({(1st𝑦)} × 𝑤))
3029eleq2d 2823 . . . . . . . . 9 (𝑧 = {(1st𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st𝑦)} × 𝑤)))
3129sseq1d 3975 . . . . . . . . 9 (𝑧 = {(1st𝑦)} → ((𝑧 × 𝑤) ⊆ 𝑥 ↔ ({(1st𝑦)} × 𝑤) ⊆ 𝑥))
3230, 31anbi12d 631 . . . . . . . 8 (𝑧 = {(1st𝑦)} → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ ({(1st𝑦)} × 𝑤) ∧ ({(1st𝑦)} × 𝑤) ⊆ 𝑥)))
33 xpeq2 5654 . . . . . . . . . . 11 (𝑤 = {(2nd𝑦)} → ({(1st𝑦)} × 𝑤) = ({(1st𝑦)} × {(2nd𝑦)}))
34 fvex 6855 . . . . . . . . . . . 12 (1st𝑦) ∈ V
35 fvex 6855 . . . . . . . . . . . 12 (2nd𝑦) ∈ V
3634, 35xpsn 7087 . . . . . . . . . . 11 ({(1st𝑦)} × {(2nd𝑦)}) = {⟨(1st𝑦), (2nd𝑦)⟩}
3733, 36eqtrdi 2792 . . . . . . . . . 10 (𝑤 = {(2nd𝑦)} → ({(1st𝑦)} × 𝑤) = {⟨(1st𝑦), (2nd𝑦)⟩})
3837eleq2d 2823 . . . . . . . . 9 (𝑤 = {(2nd𝑦)} → (𝑦 ∈ ({(1st𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩}))
3937sseq1d 3975 . . . . . . . . 9 (𝑤 = {(2nd𝑦)} → (({(1st𝑦)} × 𝑤) ⊆ 𝑥 ↔ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥))
4038, 39anbi12d 631 . . . . . . . 8 (𝑤 = {(2nd𝑦)} → ((𝑦 ∈ ({(1st𝑦)} × 𝑤) ∧ ({(1st𝑦)} × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩} ∧ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)))
4132, 40rspc2ev 3592 . . . . . . 7 (({(1st𝑦)} ∈ 𝒫 𝐴 ∧ {(2nd𝑦)} ∈ 𝒫 𝐵 ∧ (𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩} ∧ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
4217, 20, 25, 28, 41syl112anc 1374 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
4342expr 457 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝑦𝑥) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
4443ralrimdva 3151 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
45 eltx 22919 . . . . 5 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵) ↔ ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
461, 2, 45syl2an 596 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵) ↔ ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
4744, 46sylibrd 258 . . 3 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → 𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵)))
4847ssrdv 3950 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
4912, 48eqssd 3961 1 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3064  wrex 3073  wss 3910  𝒫 cpw 4560  {csn 4586  cop 4592   cuni 4865   × cxp 5631  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  Topctop 22242   ×t ctx 22911
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-topgen 17325  df-top 22243  df-topon 22260  df-bases 22296  df-tx 22913
This theorem is referenced by:  distgp  23450
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