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Theorem ptuni 23067
Description: The base set for the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypothesis
Ref Expression
ptuni.1 𝐽 = (∏t𝐹)
Assertion
Ref Expression
ptuni ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = 𝐽)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑉
Allowed substitution hint:   𝐽(𝑥)

Proof of Theorem ptuni
Dummy variables 𝑔 𝑘 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . 4 {𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))} = {𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}
21ptbas 23052 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
3 unitg 22439 . . 3 ({𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases → (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}) = {𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))})
42, 3syl 17 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}) = {𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))})
5 ptuni.1 . . . 4 𝐽 = (∏t𝐹)
6 ffn 6707 . . . . 5 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
71ptval 23043 . . . . 5 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
86, 7sylan2 594 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
95, 8eqtrid 2785 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
109unieqd 4918 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
111ptuni2 23049 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = {𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))})
124, 10, 113eqtr4rd 2784 1 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3062  wrex 3071  cdif 3943   cuni 4904   Fn wfn 6530  wf 6531  cfv 6535  Xcixp 8879  Fincfn 8927  topGenctg 17370  tcpt 17371  Topctop 22364  TopBasesctb 22417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3965  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-int 4947  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6359  df-on 6360  df-lim 6361  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-om 7843  df-1o 8453  df-er 8691  df-ixp 8880  df-en 8928  df-fin 8931  df-fi 9393  df-topgen 17376  df-pt 17377  df-top 22365  df-bases 22418
This theorem is referenced by:  ptunimpt  23068  ptval2  23074  ptpjcn  23084  ptcld  23086  ptcn  23100  pthaus  23111  ptrescn  23112  ptuncnv  23280  ptunhmeo  23281  ptcmpfi  23286  ptcmplem1  23525  ptcmpg  23530  ptpconn  34155  ptrest  36392
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