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Theorem ptopn 23531
Description: A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
ptopn.1 (𝜑𝐴𝑉)
ptopn.2 (𝜑𝐹:𝐴⟶Top)
ptopn.3 (𝜑𝑊 ∈ Fin)
ptopn.4 ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))
ptopn.5 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))
Assertion
Ref Expression
ptopn (𝜑X𝑘𝐴 𝑆 ∈ (∏t𝐹))
Distinct variable groups:   𝐴,𝑘   𝑘,𝐹   𝑘,𝑉   𝜑,𝑘   𝑘,𝑊
Allowed substitution hint:   𝑆(𝑘)

Proof of Theorem ptopn
Dummy variables 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ptopn.1 . . . . 5 (𝜑𝐴𝑉)
2 ptopn.2 . . . . 5 (𝜑𝐹:𝐴⟶Top)
3 eqid 2725 . . . . . 6 {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
43ptbas 23527 . . . . 5 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
51, 2, 4syl2anc 582 . . . 4 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
6 bastg 22913 . . . 4 ({𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ⊆ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
75, 6syl 17 . . 3 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ⊆ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
82ffnd 6724 . . . 4 (𝜑𝐹 Fn 𝐴)
93ptval 23518 . . . 4 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
101, 8, 9syl2anc 582 . . 3 (𝜑 → (∏t𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}))
117, 10sseqtrrd 4018 . 2 (𝜑 → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ⊆ (∏t𝐹))
12 ptopn.3 . . 3 (𝜑𝑊 ∈ Fin)
13 ptopn.4 . . 3 ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))
14 ptopn.5 . . 3 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))
153, 1, 12, 13, 14elptr2 23522 . 2 (𝜑X𝑘𝐴 𝑆 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))})
1611, 15sseldd 3977 1 (𝜑X𝑘𝐴 𝑆 ∈ (∏t𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wral 3050  wrex 3059  cdif 3941  wss 3944   cuni 4909   Fn wfn 6544  wf 6545  cfv 6549  Xcixp 8916  Fincfn 8964  topGenctg 17422  tcpt 17423  Topctop 22839  TopBasesctb 22892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-om 7872  df-1o 8487  df-er 8725  df-ixp 8917  df-en 8965  df-fin 8968  df-fi 9436  df-topgen 17428  df-pt 17429  df-top 22840  df-bases 22893
This theorem is referenced by:  ptopn2  23532  xkopt  23603  poimirlem30  37254
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