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Theorem ptuni2 22755
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 22754 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ∈ 𝐵)
3 elssuni 4874 . . 3 (X𝑘𝐴 (𝐹𝑘) ∈ 𝐵X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
42, 3syl 17 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
5 simpr2 1193 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦))
6 elssuni 4874 . . . . . . . . . . 11 ((𝑔𝑦) ∈ (𝐹𝑦) → (𝑔𝑦) ⊆ (𝐹𝑦))
76ralimi 3080 . . . . . . . . . 10 (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) → ∀𝑦𝐴 (𝑔𝑦) ⊆ (𝐹𝑦))
8 ss2ixp 8718 . . . . . . . . . 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 6792 . . . . . . . . . . 11 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
1110unieqd 4855 . . . . . . . . . 10 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
1211cbvixpv 8723 . . . . . . . . 9 X𝑦𝐴 (𝐹𝑦) = X𝑘𝐴 (𝐹𝑘)
139, 12sseqtrdi 3973 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘))
14 velpw 4541 . . . . . . . . 9 (𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝑥X𝑘𝐴 (𝐹𝑘))
15 sseq1 3948 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥X𝑘𝐴 (𝐹𝑘) ↔ X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘)))
1614, 15bitrid 282 . . . . . . . 8 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘)))
1713, 16syl5ibrcom 246 . . . . . . 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 4004 . . . 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 3971 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘))
22 sspwuni 5032 . . 3 (𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝐵X𝑘𝐴 (𝐹𝑘))
2321, 22sylib 217 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵X𝑘𝐴 (𝐹𝑘))
244, 23eqssd 3940 1 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1085   = wceq 1537  wex 1777  wcel 2101  {cab 2710  wral 3059  wrex 3068  cdif 3886  wss 3889  𝒫 cpw 4536   cuni 4841   Fn wfn 6442  wf 6443  cfv 6447  Xcixp 8705  Fincfn 8753  Topctop 22070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-rep 5212  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3223  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3908  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-iun 4929  df-br 5078  df-opab 5140  df-mpt 5161  df-tr 5195  df-id 5491  df-eprel 5497  df-po 5505  df-so 5506  df-fr 5546  df-we 5548  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-om 7733  df-ixp 8706  df-en 8754  df-fin 8757  df-top 22071
This theorem is referenced by:  ptbasin2  22757  ptbasfi  22760  ptuni  22773
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