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Theorem txdis 21345
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 20710 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 distop 20710 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ∈ Top)
3 unipw 4879 . . . . . . 7 𝒫 𝐴 = 𝐴
43eqcomi 2630 . . . . . 6 𝐴 = 𝒫 𝐴
5 unipw 4879 . . . . . . 7 𝒫 𝐵 = 𝐵
65eqcomi 2630 . . . . . 6 𝐵 = 𝒫 𝐵
74, 6txuni 21305 . . . . 5 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝐴 × 𝐵) = (𝒫 𝐴 ×t 𝒫 𝐵))
81, 2, 7syl2an 494 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) = (𝒫 𝐴 ×t 𝒫 𝐵))
9 eqimss2 3637 . . . 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 4577 . . 3 ((𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵) ↔ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
1210, 11sylibr 224 . 2 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵))
13 elelpwi 4142 . . . . . . . . 9 ((𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵)) → 𝑦 ∈ (𝐴 × 𝐵))
1413adantl 482 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ (𝐴 × 𝐵))
15 xp1st 7143 . . . . . . . 8 (𝑦 ∈ (𝐴 × 𝐵) → (1st𝑦) ∈ 𝐴)
16 snelpwi 4873 . . . . . . . 8 ((1st𝑦) ∈ 𝐴 → {(1st𝑦)} ∈ 𝒫 𝐴)
1714, 15, 163syl 18 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(1st𝑦)} ∈ 𝒫 𝐴)
18 xp2nd 7144 . . . . . . . 8 (𝑦 ∈ (𝐴 × 𝐵) → (2nd𝑦) ∈ 𝐵)
19 snelpwi 4873 . . . . . . . 8 ((2nd𝑦) ∈ 𝐵 → {(2nd𝑦)} ∈ 𝒫 𝐵)
2014, 18, 193syl 18 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(2nd𝑦)} ∈ 𝒫 𝐵)
21 vsnid 4180 . . . . . . . 8 𝑦 ∈ {𝑦}
22 1st2nd2 7150 . . . . . . . . . 10 (𝑦 ∈ (𝐴 × 𝐵) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2314, 22syl 17 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2423sneqd 4160 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st𝑦), (2nd𝑦)⟩})
2521, 24syl5eleq 2704 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩})
26 simprl 793 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦𝑥)
2723, 26eqeltrrd 2699 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝑥)
2827snssd 4309 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)
29 xpeq1 5088 . . . . . . . . . 10 (𝑧 = {(1st𝑦)} → (𝑧 × 𝑤) = ({(1st𝑦)} × 𝑤))
3029eleq2d 2684 . . . . . . . . 9 (𝑧 = {(1st𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st𝑦)} × 𝑤)))
3129sseq1d 3611 . . . . . . . . 9 (𝑧 = {(1st𝑦)} → ((𝑧 × 𝑤) ⊆ 𝑥 ↔ ({(1st𝑦)} × 𝑤) ⊆ 𝑥))
3230, 31anbi12d 746 . . . . . . . 8 (𝑧 = {(1st𝑦)} → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ ({(1st𝑦)} × 𝑤) ∧ ({(1st𝑦)} × 𝑤) ⊆ 𝑥)))
33 xpeq2 5089 . . . . . . . . . . 11 (𝑤 = {(2nd𝑦)} → ({(1st𝑦)} × 𝑤) = ({(1st𝑦)} × {(2nd𝑦)}))
34 fvex 6158 . . . . . . . . . . . 12 (1st𝑦) ∈ V
35 fvex 6158 . . . . . . . . . . . 12 (2nd𝑦) ∈ V
3634, 35xpsn 6361 . . . . . . . . . . 11 ({(1st𝑦)} × {(2nd𝑦)}) = {⟨(1st𝑦), (2nd𝑦)⟩}
3733, 36syl6eq 2671 . . . . . . . . . 10 (𝑤 = {(2nd𝑦)} → ({(1st𝑦)} × 𝑤) = {⟨(1st𝑦), (2nd𝑦)⟩})
3837eleq2d 2684 . . . . . . . . 9 (𝑤 = {(2nd𝑦)} → (𝑦 ∈ ({(1st𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩}))
3937sseq1d 3611 . . . . . . . . 9 (𝑤 = {(2nd𝑦)} → (({(1st𝑦)} × 𝑤) ⊆ 𝑥 ↔ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥))
4038, 39anbi12d 746 . . . . . . . 8 (𝑤 = {(2nd𝑦)} → ((𝑦 ∈ ({(1st𝑦)} × 𝑤) ∧ ({(1st𝑦)} × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩} ∧ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)))
4132, 40rspc2ev 3308 . . . . . . 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 1327 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
4342expr 642 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝑦𝑥) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
4443ralrimdva 2963 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
45 eltx 21281 . . . . 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 494 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵) ↔ ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
4744, 46sylibrd 249 . . 3 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → 𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵)))
4847ssrdv 3589 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
4912, 48eqssd 3600 1 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  wss 3555  𝒫 cpw 4130  {csn 4148  cop 4154   cuni 4402   × cxp 5072  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  Topctop 20617   ×t ctx 21273
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-topgen 16025  df-top 20621  df-bases 20622  df-topon 20623  df-tx 21275
This theorem is referenced by:  distgp  21813
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