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Theorem txdis 21649
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 21013 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 distop 21013 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ∈ Top)
3 unipw 5108 . . . . . . 7 𝒫 𝐴 = 𝐴
43eqcomi 2815 . . . . . 6 𝐴 = 𝒫 𝐴
5 unipw 5108 . . . . . . 7 𝒫 𝐵 = 𝐵
65eqcomi 2815 . . . . . 6 𝐵 = 𝒫 𝐵
74, 6txuni 21609 . . . . 5 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝐴 × 𝐵) = (𝒫 𝐴 ×t 𝒫 𝐵))
81, 2, 7syl2an 585 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) = (𝒫 𝐴 ×t 𝒫 𝐵))
9 eqimss2 3855 . . . 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 4803 . . 3 ((𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵) ↔ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
1210, 11sylibr 225 . 2 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵))
13 elelpwi 4364 . . . . . . . . 9 ((𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵)) → 𝑦 ∈ (𝐴 × 𝐵))
1413adantl 469 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ (𝐴 × 𝐵))
15 xp1st 7430 . . . . . . . 8 (𝑦 ∈ (𝐴 × 𝐵) → (1st𝑦) ∈ 𝐴)
16 snelpwi 5102 . . . . . . . 8 ((1st𝑦) ∈ 𝐴 → {(1st𝑦)} ∈ 𝒫 𝐴)
1714, 15, 163syl 18 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(1st𝑦)} ∈ 𝒫 𝐴)
18 xp2nd 7431 . . . . . . . 8 (𝑦 ∈ (𝐴 × 𝐵) → (2nd𝑦) ∈ 𝐵)
19 snelpwi 5102 . . . . . . . 8 ((2nd𝑦) ∈ 𝐵 → {(2nd𝑦)} ∈ 𝒫 𝐵)
2014, 18, 193syl 18 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(2nd𝑦)} ∈ 𝒫 𝐵)
21 vsnid 4403 . . . . . . . 8 𝑦 ∈ {𝑦}
22 1st2nd2 7437 . . . . . . . . . 10 (𝑦 ∈ (𝐴 × 𝐵) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2314, 22syl 17 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2423sneqd 4382 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st𝑦), (2nd𝑦)⟩})
2521, 24syl5eleq 2891 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩})
26 simprl 778 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦𝑥)
2723, 26eqeltrrd 2886 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝑥)
2827snssd 4530 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)
29 xpeq1 5325 . . . . . . . . . 10 (𝑧 = {(1st𝑦)} → (𝑧 × 𝑤) = ({(1st𝑦)} × 𝑤))
3029eleq2d 2871 . . . . . . . . 9 (𝑧 = {(1st𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st𝑦)} × 𝑤)))
3129sseq1d 3829 . . . . . . . . 9 (𝑧 = {(1st𝑦)} → ((𝑧 × 𝑤) ⊆ 𝑥 ↔ ({(1st𝑦)} × 𝑤) ⊆ 𝑥))
3230, 31anbi12d 618 . . . . . . . 8 (𝑧 = {(1st𝑦)} → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ ({(1st𝑦)} × 𝑤) ∧ ({(1st𝑦)} × 𝑤) ⊆ 𝑥)))
33 xpeq2 5331 . . . . . . . . . . 11 (𝑤 = {(2nd𝑦)} → ({(1st𝑦)} × 𝑤) = ({(1st𝑦)} × {(2nd𝑦)}))
34 fvex 6421 . . . . . . . . . . . 12 (1st𝑦) ∈ V
35 fvex 6421 . . . . . . . . . . . 12 (2nd𝑦) ∈ V
3634, 35xpsn 6630 . . . . . . . . . . 11 ({(1st𝑦)} × {(2nd𝑦)}) = {⟨(1st𝑦), (2nd𝑦)⟩}
3733, 36syl6eq 2856 . . . . . . . . . 10 (𝑤 = {(2nd𝑦)} → ({(1st𝑦)} × 𝑤) = {⟨(1st𝑦), (2nd𝑦)⟩})
3837eleq2d 2871 . . . . . . . . 9 (𝑤 = {(2nd𝑦)} → (𝑦 ∈ ({(1st𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩}))
3937sseq1d 3829 . . . . . . . . 9 (𝑤 = {(2nd𝑦)} → (({(1st𝑦)} × 𝑤) ⊆ 𝑥 ↔ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥))
4038, 39anbi12d 618 . . . . . . . 8 (𝑤 = {(2nd𝑦)} → ((𝑦 ∈ ({(1st𝑦)} × 𝑤) ∧ ({(1st𝑦)} × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩} ∧ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)))
4132, 40rspc2ev 3517 . . . . . . 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 1486 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
4342expr 446 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝑦𝑥) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
4443ralrimdva 3157 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
45 eltx 21585 . . . . 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 585 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵) ↔ ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
4744, 46sylibrd 250 . . 3 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → 𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵)))
4847ssrdv 3804 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
4912, 48eqssd 3815 1 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wcel 2156  wral 3096  wrex 3097  wss 3769  𝒫 cpw 4351  {csn 4370  cop 4376   cuni 4630   × cxp 5309  cfv 6101  (class class class)co 6874  1st c1st 7396  2nd c2nd 7397  Topctop 20911   ×t ctx 21577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-1st 7398  df-2nd 7399  df-topgen 16309  df-top 20912  df-topon 20929  df-bases 20964  df-tx 21579
This theorem is referenced by:  distgp  22116
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