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Theorem txdis 13444
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 13252 . . . . 5 (𝐴𝑉 → 𝒫 𝐴 ∈ Top)
2 distop 13252 . . . . 5 (𝐵𝑊 → 𝒫 𝐵 ∈ Top)
3 unipw 4214 . . . . . . 7 𝒫 𝐴 = 𝐴
43eqcomi 2181 . . . . . 6 𝐴 = 𝒫 𝐴
5 unipw 4214 . . . . . . 7 𝒫 𝐵 = 𝐵
65eqcomi 2181 . . . . . 6 𝐵 = 𝒫 𝐵
74, 6txuni 13430 . . . . 5 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝐴 × 𝐵) = (𝒫 𝐴 ×t 𝒫 𝐵))
81, 2, 7syl2an 289 . . . 4 ((𝐴𝑉𝐵𝑊) → (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))
108, 9syl 14 . . 3 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
11 sspwuni 3968 . . 3 ((𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵) ↔ (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ (𝐴 × 𝐵))
1210, 11sylibr 134 . 2 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) ⊆ 𝒫 (𝐴 × 𝐵))
13 elelpwi 3586 . . . . . . . . 9 ((𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵)) → 𝑦 ∈ (𝐴 × 𝐵))
1413adantl 277 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ (𝐴 × 𝐵))
15 xp1st 6160 . . . . . . . 8 (𝑦 ∈ (𝐴 × 𝐵) → (1st𝑦) ∈ 𝐴)
16 snelpwi 4209 . . . . . . . 8 ((1st𝑦) ∈ 𝐴 → {(1st𝑦)} ∈ 𝒫 𝐴)
1714, 15, 163syl 17 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(1st𝑦)} ∈ 𝒫 𝐴)
18 xp2nd 6161 . . . . . . . 8 (𝑦 ∈ (𝐴 × 𝐵) → (2nd𝑦) ∈ 𝐵)
19 snelpwi 4209 . . . . . . . 8 ((2nd𝑦) ∈ 𝐵 → {(2nd𝑦)} ∈ 𝒫 𝐵)
2014, 18, 193syl 17 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {(2nd𝑦)} ∈ 𝒫 𝐵)
21 vsnid 3623 . . . . . . . 8 𝑦 ∈ {𝑦}
22 1st2nd2 6170 . . . . . . . . . 10 (𝑦 ∈ (𝐴 × 𝐵) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2314, 22syl 14 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
2423sneqd 3604 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {𝑦} = {⟨(1st𝑦), (2nd𝑦)⟩})
2521, 24eleqtrid 2266 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩})
26 simprl 529 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → 𝑦𝑥)
2723, 26eqeltrrd 2255 . . . . . . . 8 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ⟨(1st𝑦), (2nd𝑦)⟩ ∈ 𝑥)
2827snssd 3736 . . . . . . 7 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)
29 xpeq1 4637 . . . . . . . . . 10 (𝑧 = {(1st𝑦)} → (𝑧 × 𝑤) = ({(1st𝑦)} × 𝑤))
3029eleq2d 2247 . . . . . . . . 9 (𝑧 = {(1st𝑦)} → (𝑦 ∈ (𝑧 × 𝑤) ↔ 𝑦 ∈ ({(1st𝑦)} × 𝑤)))
3129sseq1d 3184 . . . . . . . . 9 (𝑧 = {(1st𝑦)} → ((𝑧 × 𝑤) ⊆ 𝑥 ↔ ({(1st𝑦)} × 𝑤) ⊆ 𝑥))
3230, 31anbi12d 473 . . . . . . . 8 (𝑧 = {(1st𝑦)} → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ ({(1st𝑦)} × 𝑤) ∧ ({(1st𝑦)} × 𝑤) ⊆ 𝑥)))
33 xpeq2 4638 . . . . . . . . . . 11 (𝑤 = {(2nd𝑦)} → ({(1st𝑦)} × 𝑤) = ({(1st𝑦)} × {(2nd𝑦)}))
34 1stexg 6162 . . . . . . . . . . . . 13 (𝑦 ∈ V → (1st𝑦) ∈ V)
3534elv 2741 . . . . . . . . . . . 12 (1st𝑦) ∈ V
36 2ndexg 6163 . . . . . . . . . . . . 13 (𝑦 ∈ V → (2nd𝑦) ∈ V)
3736elv 2741 . . . . . . . . . . . 12 (2nd𝑦) ∈ V
3835, 37xpsn 5688 . . . . . . . . . . 11 ({(1st𝑦)} × {(2nd𝑦)}) = {⟨(1st𝑦), (2nd𝑦)⟩}
3933, 38eqtrdi 2226 . . . . . . . . . 10 (𝑤 = {(2nd𝑦)} → ({(1st𝑦)} × 𝑤) = {⟨(1st𝑦), (2nd𝑦)⟩})
4039eleq2d 2247 . . . . . . . . 9 (𝑤 = {(2nd𝑦)} → (𝑦 ∈ ({(1st𝑦)} × 𝑤) ↔ 𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩}))
4139sseq1d 3184 . . . . . . . . 9 (𝑤 = {(2nd𝑦)} → (({(1st𝑦)} × 𝑤) ⊆ 𝑥 ↔ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥))
4240, 41anbi12d 473 . . . . . . . 8 (𝑤 = {(2nd𝑦)} → ((𝑦 ∈ ({(1st𝑦)} × 𝑤) ∧ ({(1st𝑦)} × 𝑤) ⊆ 𝑥) ↔ (𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩} ∧ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)))
4332, 42rspc2ev 2856 . . . . . . 7 (({(1st𝑦)} ∈ 𝒫 𝐴 ∧ {(2nd𝑦)} ∈ 𝒫 𝐵 ∧ (𝑦 ∈ {⟨(1st𝑦), (2nd𝑦)⟩} ∧ {⟨(1st𝑦), (2nd𝑦)⟩} ⊆ 𝑥)) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
4417, 20, 25, 28, 43syl112anc 1242 . . . . . 6 (((𝐴𝑉𝐵𝑊) ∧ (𝑦𝑥𝑥 ∈ 𝒫 (𝐴 × 𝐵))) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
4544expr 375 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ 𝑦𝑥) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∃𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
4645ralrimdva 2557 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
47 eltx 13426 . . . . 5 ((𝒫 𝐴 ∈ Top ∧ 𝒫 𝐵 ∈ Top) → (𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵) ↔ ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
481, 2, 47syl2an 289 . . . 4 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵) ↔ ∀𝑦𝑥𝑧 ∈ 𝒫 𝐴𝑤 ∈ 𝒫 𝐵(𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
4946, 48sylibrd 169 . . 3 ((𝐴𝑉𝐵𝑊) → (𝑥 ∈ 𝒫 (𝐴 × 𝐵) → 𝑥 ∈ (𝒫 𝐴 ×t 𝒫 𝐵)))
5049ssrdv 3161 . 2 ((𝐴𝑉𝐵𝑊) → 𝒫 (𝐴 × 𝐵) ⊆ (𝒫 𝐴 ×t 𝒫 𝐵))
5112, 50eqssd 3172 1 ((𝐴𝑉𝐵𝑊) → (𝒫 𝐴 ×t 𝒫 𝐵) = 𝒫 (𝐴 × 𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1353  wcel 2148  wral 2455  wrex 2456  Vcvv 2737  wss 3129  𝒫 cpw 3574  {csn 3591  cop 3594   cuni 3807   × cxp 4621  cfv 5212  (class class class)co 5869  1st c1st 6133  2nd c2nd 6134  Topctop 13162   ×t ctx 13419
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 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
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 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-topgen 12657  df-top 13163  df-topon 13176  df-bases 13208  df-tx 13420
This theorem is referenced by: (None)
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