MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elptr2 Structured version   Visualization version   GIF version

Theorem elptr2 23078
Description: A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypotheses
Ref Expression
ptbas.1 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
elptr2.1 (𝜑𝐴𝑉)
elptr2.2 (𝜑𝑊 ∈ Fin)
elptr2.3 ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))
elptr2.4 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))
Assertion
Ref Expression
elptr2 (𝜑X𝑘𝐴 𝑆𝐵)
Distinct variable groups:   𝐵,𝑘   𝑥,𝑔,𝑦   𝜑,𝑘   𝑔,𝑘,𝑧,𝐴,𝑥,𝑦   𝑔,𝐹,𝑘,𝑥,𝑦,𝑧   𝑆,𝑔,𝑥   𝑔,𝑉,𝑘,𝑥,𝑦,𝑧   𝑘,𝑊,𝑦   𝑦,𝑆
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑔)   𝐵(𝑥,𝑦,𝑧,𝑔)   𝑆(𝑧,𝑘)   𝑊(𝑥,𝑧,𝑔)

Proof of Theorem elptr2
StepHypRef Expression
1 nffvmpt1 6903 . . . 4 𝑘((𝑘𝐴𝑆)‘𝑦)
2 nfcv 2904 . . . 4 𝑦((𝑘𝐴𝑆)‘𝑘)
3 fveq2 6892 . . . 4 (𝑦 = 𝑘 → ((𝑘𝐴𝑆)‘𝑦) = ((𝑘𝐴𝑆)‘𝑘))
41, 2, 3cbvixp 8908 . . 3 X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘)
5 simpr 486 . . . . 5 ((𝜑𝑘𝐴) → 𝑘𝐴)
6 elptr2.3 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))
7 eqid 2733 . . . . . 6 (𝑘𝐴𝑆) = (𝑘𝐴𝑆)
87fvmpt2 7010 . . . . 5 ((𝑘𝐴𝑆 ∈ (𝐹𝑘)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
95, 6, 8syl2anc 585 . . . 4 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
109ixpeq2dva 8906 . . 3 (𝜑X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) = X𝑘𝐴 𝑆)
114, 10eqtrid 2785 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 𝑆)
12 elptr2.1 . . 3 (𝜑𝐴𝑉)
136ralrimiva 3147 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ∈ (𝐹𝑘))
147fnmpt 6691 . . . 4 (∀𝑘𝐴 𝑆 ∈ (𝐹𝑘) → (𝑘𝐴𝑆) Fn 𝐴)
1513, 14syl 17 . . 3 (𝜑 → (𝑘𝐴𝑆) Fn 𝐴)
169, 6eqeltrd 2834 . . . . 5 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
1716ralrimiva 3147 . . . 4 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
181nfel1 2920 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦)
19 nfv 1918 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)
20 fveq2 6892 . . . . . 6 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
213, 20eleq12d 2828 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)))
2218, 19, 21cbvralw 3304 . . . 4 (∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
2317, 22sylibr 233 . . 3 (𝜑 → ∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦))
24 elptr2.2 . . 3 (𝜑𝑊 ∈ Fin)
25 eldifi 4127 . . . . . . 7 (𝑘 ∈ (𝐴𝑊) → 𝑘𝐴)
2625, 9sylan2 594 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
27 elptr2.4 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))
2826, 27eqtrd 2773 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
2928ralrimiva 3147 . . . 4 (𝜑 → ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
301nfeq1 2919 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦)
31 nfv 1918 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)
3220unieqd 4923 . . . . . 6 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
333, 32eqeq12d 2749 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)))
3430, 31, 33cbvralw 3304 . . . 4 (∀𝑦 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
3529, 34sylibr 233 . . 3 (𝜑 → ∀𝑦 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦))
36 ptbas.1 . . . 4 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
3736elptr 23077 . . 3 ((𝐴𝑉 ∧ ((𝑘𝐴𝑆) Fn 𝐴 ∧ ∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦))) → X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ 𝐵)
3812, 15, 23, 24, 35, 37syl122anc 1380 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ 𝐵)
3911, 38eqeltrrd 2835 1 (𝜑X𝑘𝐴 𝑆𝐵)
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 3946   cuni 4909  cmpt 5232   Fn wfn 6539  cfv 6544  Xcixp 8891  Fincfn 8939
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ixp 8892
This theorem is referenced by:  ptbasid  23079  ptbasin  23081  ptpjpre2  23084  ptopn  23087
  Copyright terms: Public domain W3C validator