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Theorem ptuni 22294
 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 2758 . . . 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 22279 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
3 unitg 21667 . . 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 6498 . . . . 5 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
71ptval 22270 . . . . 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 595 . . . 4 ((𝐴𝑉𝐹:𝐴⟶Top) → (∏t𝐹) = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
95, 8syl5eq 2805 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
109unieqd 4812 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
111ptuni2 22276 . 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 2804 1 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = 𝐽)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735  ∀wral 3070  ∃wrex 3071   ∖ cdif 3855  ∪ cuni 4798   Fn wfn 6330  ⟶wf 6331  ‘cfv 6335  Xcixp 8479  Fincfn 8527  topGenctg 16769  ∏tcpt 16770  Topctop 21593  TopBasesctb 21645 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-int 4839  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-om 7580  df-1o 8112  df-er 8299  df-ixp 8480  df-en 8528  df-fin 8531  df-fi 8908  df-topgen 16775  df-pt 16776  df-top 21594  df-bases 21646 This theorem is referenced by:  ptunimpt  22295  ptval2  22301  ptpjcn  22311  ptcld  22313  ptcn  22327  pthaus  22338  ptrescn  22339  ptuncnv  22507  ptunhmeo  22508  ptcmpfi  22513  ptcmplem1  22752  ptcmpg  22757  ptpconn  32711  ptrest  35336
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