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Theorem ptuni2 23585
Description: The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypothesis
Ref Expression
ptbas.1 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
Assertion
Ref Expression
ptuni2 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐵)
Distinct variable groups:   𝐵,𝑘   𝑥,𝑔,𝑦,𝑘,𝑧,𝐴   𝑔,𝐹,𝑘,𝑥,𝑦,𝑧   𝑔,𝑉,𝑘,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑔)

Proof of Theorem ptuni2
StepHypRef Expression
1 ptbas.1 . . . 4 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
21ptbasid 23584 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ∈ 𝐵)
3 elssuni 4936 . . 3 (X𝑘𝐴 (𝐹𝑘) ∈ 𝐵X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
42, 3syl 17 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
5 simpr2 1195 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦))
6 elssuni 4936 . . . . . . . . . . 11 ((𝑔𝑦) ∈ (𝐹𝑦) → (𝑔𝑦) ⊆ (𝐹𝑦))
76ralimi 3082 . . . . . . . . . 10 (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) → ∀𝑦𝐴 (𝑔𝑦) ⊆ (𝐹𝑦))
8 ss2ixp 8951 . . . . . . . . . 10 (∀𝑦𝐴 (𝑔𝑦) ⊆ (𝐹𝑦) → X𝑦𝐴 (𝑔𝑦) ⊆ X𝑦𝐴 (𝐹𝑦))
95, 7, 83syl 18 . . . . . . . . 9 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑔𝑦) ⊆ X𝑦𝐴 (𝐹𝑦))
10 fveq2 6905 . . . . . . . . . . 11 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
1110unieqd 4919 . . . . . . . . . 10 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
1211cbvixpv 8956 . . . . . . . . 9 X𝑦𝐴 (𝐹𝑦) = X𝑘𝐴 (𝐹𝑘)
139, 12sseqtrdi 4023 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘))
14 velpw 4604 . . . . . . . . 9 (𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝑥X𝑘𝐴 (𝐹𝑘))
15 sseq1 4008 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥X𝑘𝐴 (𝐹𝑘) ↔ X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘)))
1614, 15bitrid 283 . . . . . . . 8 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘)))
1713, 16syl5ibrcom 247 . . . . . . 7 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → (𝑥 = X𝑦𝐴 (𝑔𝑦) → 𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘)))
1817expimpd 453 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) → 𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘)))
1918exlimdv 1932 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) → 𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘)))
2019abssdv 4067 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘))
211, 20eqsstrid 4021 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘))
22 sspwuni 5099 . . 3 (𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝐵X𝑘𝐴 (𝐹𝑘))
2321, 22sylib 218 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵X𝑘𝐴 (𝐹𝑘))
244, 23eqssd 4000 1 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  wrex 3069  cdif 3947  wss 3950  𝒫 cpw 4599   cuni 4906   Fn wfn 6555  wf 6556  cfv 6560  Xcixp 8938  Fincfn 8986  Topctop 22900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-ord 6386  df-on 6387  df-lim 6388  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-om 7889  df-ixp 8939  df-en 8987  df-fin 8990  df-top 22901
This theorem is referenced by:  ptbasin2  23587  ptbasfi  23590  ptuni  23603
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