MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ptuni2 Structured version   Visualization version   GIF version

Theorem ptuni2 21302
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 21301 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ∈ 𝐵)
3 elssuni 4438 . . 3 (X𝑘𝐴 (𝐹𝑘) ∈ 𝐵X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
42, 3syl 17 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
5 simpr2 1066 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦))
6 elssuni 4438 . . . . . . . . . . 11 ((𝑔𝑦) ∈ (𝐹𝑦) → (𝑔𝑦) ⊆ (𝐹𝑦))
76ralimi 2947 . . . . . . . . . 10 (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) → ∀𝑦𝐴 (𝑔𝑦) ⊆ (𝐹𝑦))
8 ss2ixp 7873 . . . . . . . . . 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 6153 . . . . . . . . . . 11 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
1110unieqd 4417 . . . . . . . . . 10 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
1211cbvixpv 7878 . . . . . . . . 9 X𝑦𝐴 (𝐹𝑦) = X𝑘𝐴 (𝐹𝑘)
139, 12syl6sseq 3635 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘))
14 selpw 4142 . . . . . . . . 9 (𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝑥X𝑘𝐴 (𝐹𝑘))
15 sseq1 3610 . . . . . . . . 9 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥X𝑘𝐴 (𝐹𝑘) ↔ X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘)))
1614, 15syl5bb 272 . . . . . . . 8 (𝑥 = X𝑦𝐴 (𝑔𝑦) → (𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘)))
1713, 16syl5ibrcom 237 . . . . . . 7 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → (𝑥 = X𝑦𝐴 (𝑔𝑦) → 𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘)))
1817expimpd 628 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) → 𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘)))
1918exlimdv 1858 . . . . 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 3660 . . . 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, 20syl5eqss 3633 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘))
22 sspwuni 4582 . . 3 (𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝐵X𝑘𝐴 (𝐹𝑘))
2321, 22sylib 208 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵X𝑘𝐴 (𝐹𝑘))
244, 23eqssd 3604 1 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wral 2907  wrex 2908  cdif 3556  wss 3559  𝒫 cpw 4135   cuni 4407   Fn wfn 5847  wf 5848  cfv 5852  Xcixp 7860  Fincfn 7907  Topctop 20630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-om 7020  df-ixp 7861  df-en 7908  df-fin 7911  df-top 20631
This theorem is referenced by:  ptbasin2  21304  ptbasfi  21307  ptuni  21320
  Copyright terms: Public domain W3C validator