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Theorem ptuni2 23600
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 23599 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ∈ 𝐵)
3 elssuni 4942 . . 3 (X𝑘𝐴 (𝐹𝑘) ∈ 𝐵X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
42, 3syl 17 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
5 simpr2 1194 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦))
6 elssuni 4942 . . . . . . . . . . 11 ((𝑔𝑦) ∈ (𝐹𝑦) → (𝑔𝑦) ⊆ (𝐹𝑦))
76ralimi 3081 . . . . . . . . . 10 (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) → ∀𝑦𝐴 (𝑔𝑦) ⊆ (𝐹𝑦))
8 ss2ixp 8949 . . . . . . . . . 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 6907 . . . . . . . . . . 11 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
1110unieqd 4925 . . . . . . . . . 10 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
1211cbvixpv 8954 . . . . . . . . 9 X𝑦𝐴 (𝐹𝑦) = X𝑘𝐴 (𝐹𝑘)
139, 12sseqtrdi 4046 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘))
14 velpw 4610 . . . . . . . . 9 (𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝑥X𝑘𝐴 (𝐹𝑘))
15 sseq1 4021 . . . . . . . . 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 1931 . . . . 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 4078 . . . 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 4044 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘))
22 sspwuni 5105 . . 3 (𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝐵X𝑘𝐴 (𝐹𝑘))
2321, 22sylib 218 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵X𝑘𝐴 (𝐹𝑘))
244, 23eqssd 4013 1 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  cdif 3960  wss 3963  𝒫 cpw 4605   cuni 4912   Fn wfn 6558  wf 6559  cfv 6563  Xcixp 8936  Fincfn 8984  Topctop 22915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-om 7888  df-ixp 8937  df-en 8985  df-fin 8988  df-top 22916
This theorem is referenced by:  ptbasin2  23602  ptbasfi  23605  ptuni  23618
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