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

Theorem ptuni 21378
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 2620 . . . 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 21363 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → {𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))} ∈ TopBases)
3 unitg 20752 . . 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 6032 . . . . 5 (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
71ptval 21354 . . . . 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 491 . . . 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 2666 . . 3 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
109unieqd 4437 . 2 ((𝐴𝑉𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑘 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑘 = X𝑦𝐴 (𝑔𝑦))}))
111ptuni2 21360 . 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 2665 1 ((𝐴𝑉𝐹:𝐴⟶Top) → X𝑥𝐴 (𝐹𝑥) = 𝐽)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  wrex 2910  cdif 3564   cuni 4427   Fn wfn 5871  wf 5872  cfv 5876  Xcixp 7893  Fincfn 7940  topGenctg 16079  tcpt 16080  Topctop 20679  TopBasesctb 20730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-ixp 7894  df-en 7941  df-fin 7944  df-fi 8302  df-topgen 16085  df-pt 16086  df-top 20680  df-bases 20731
This theorem is referenced by:  ptunimpt  21379  ptval2  21385  ptpjcn  21395  ptcld  21397  ptcn  21411  pthaus  21422  ptrescn  21423  ptuncnv  21591  ptunhmeo  21592  ptcmpfi  21597  ptcmplem1  21837  ptcmpg  21842  ptpconn  31189  ptrest  33379
  Copyright terms: Public domain W3C validator