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Theorem elptr2 21425
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 6237 . . . 4 𝑘((𝑘𝐴𝑆)‘𝑦)
2 nfcv 2793 . . . 4 𝑦((𝑘𝐴𝑆)‘𝑘)
3 fveq2 6229 . . . 4 (𝑦 = 𝑘 → ((𝑘𝐴𝑆)‘𝑦) = ((𝑘𝐴𝑆)‘𝑘))
41, 2, 3cbvixp 7967 . . 3 X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘)
5 simpr 476 . . . . 5 ((𝜑𝑘𝐴) → 𝑘𝐴)
6 elptr2.3 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))
7 eqid 2651 . . . . . 6 (𝑘𝐴𝑆) = (𝑘𝐴𝑆)
87fvmpt2 6330 . . . . 5 ((𝑘𝐴𝑆 ∈ (𝐹𝑘)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
95, 6, 8syl2anc 694 . . . 4 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
109ixpeq2dva 7965 . . 3 (𝜑X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) = X𝑘𝐴 𝑆)
114, 10syl5eq 2697 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 𝑆)
12 elptr2.1 . . 3 (𝜑𝐴𝑉)
136ralrimiva 2995 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ∈ (𝐹𝑘))
147fnmpt 6058 . . . 4 (∀𝑘𝐴 𝑆 ∈ (𝐹𝑘) → (𝑘𝐴𝑆) Fn 𝐴)
1513, 14syl 17 . . 3 (𝜑 → (𝑘𝐴𝑆) Fn 𝐴)
169, 6eqeltrd 2730 . . . . 5 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
1716ralrimiva 2995 . . . 4 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
181nfel1 2808 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦)
19 nfv 1883 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)
20 fveq2 6229 . . . . . 6 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
213, 20eleq12d 2724 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)))
2218, 19, 21cbvral 3197 . . . 4 (∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
2317, 22sylibr 224 . . 3 (𝜑 → ∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦))
24 elptr2.2 . . 3 (𝜑𝑊 ∈ Fin)
25 eldifi 3765 . . . . . . 7 (𝑘 ∈ (𝐴𝑊) → 𝑘𝐴)
2625, 9sylan2 490 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
27 elptr2.4 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))
2826, 27eqtrd 2685 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
2928ralrimiva 2995 . . . 4 (𝜑 → ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
301nfeq1 2807 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦)
31 nfv 1883 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)
3220unieqd 4478 . . . . . 6 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
333, 32eqeq12d 2666 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)))
3430, 31, 33cbvral 3197 . . . 4 (∀𝑦 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
3529, 34sylibr 224 . . 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 21424 . . 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 1375 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ 𝐵)
3911, 38eqeltrrd 2731 1 (𝜑X𝑘𝐴 𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  cdif 3604   cuni 4468  cmpt 4762   Fn wfn 5921  cfv 5926  Xcixp 7950  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ixp 7951
This theorem is referenced by:  ptbasid  21426  ptbasin  21428  ptpjpre2  21431  ptopn  21434
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