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Theorem ptuni2 21581
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 21580 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ∈ 𝐵)
3 elssuni 4619 . . 3 (X𝑘𝐴 (𝐹𝑘) ∈ 𝐵X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
42, 3syl 17 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) ⊆ 𝐵)
5 simpr2 1236 . . . . . . . . . 10 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦))
6 elssuni 4619 . . . . . . . . . . 11 ((𝑔𝑦) ∈ (𝐹𝑦) → (𝑔𝑦) ⊆ (𝐹𝑦))
76ralimi 3090 . . . . . . . . . 10 (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) → ∀𝑦𝐴 (𝑔𝑦) ⊆ (𝐹𝑦))
8 ss2ixp 8087 . . . . . . . . . 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 6352 . . . . . . . . . . 11 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
1110unieqd 4598 . . . . . . . . . 10 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
1211cbvixpv 8092 . . . . . . . . 9 X𝑦𝐴 (𝐹𝑦) = X𝑘𝐴 (𝐹𝑘)
139, 12syl6sseq 3792 . . . . . . . 8 (((𝐴𝑉𝐹:𝐴⟶Top) ∧ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝑔𝑦) ⊆ X𝑘𝐴 (𝐹𝑘))
14 selpw 4309 . . . . . . . . 9 (𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝑥X𝑘𝐴 (𝐹𝑘))
15 sseq1 3767 . . . . . . . . 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 630 . . . . . 6 ((𝐴𝑉𝐹:𝐴⟶Top) → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) → 𝑥 ∈ 𝒫 X𝑘𝐴 (𝐹𝑘)))
1918exlimdv 2010 . . . . 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 3817 . . . 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 3790 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘))
22 sspwuni 4763 . . 3 (𝐵 ⊆ 𝒫 X𝑘𝐴 (𝐹𝑘) ↔ 𝐵X𝑘𝐴 (𝐹𝑘))
2321, 22sylib 208 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐵X𝑘𝐴 (𝐹𝑘))
244, 23eqssd 3761 1 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑘𝐴 (𝐹𝑘) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wral 3050  wrex 3051  cdif 3712  wss 3715  𝒫 cpw 4302   cuni 4588   Fn wfn 6044  wf 6045  cfv 6049  Xcixp 8074  Fincfn 8121  Topctop 20900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7231  df-ixp 8075  df-en 8122  df-fin 8125  df-top 20901
This theorem is referenced by:  ptbasin2  21583  ptbasfi  21586  ptuni  21599
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