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Theorem elptr2 21187
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 6111 . . . 4 𝑘((𝑘𝐴𝑆)‘𝑦)
2 nfcv 2751 . . . 4 𝑦((𝑘𝐴𝑆)‘𝑘)
3 fveq2 6103 . . . 4 (𝑦 = 𝑘 → ((𝑘𝐴𝑆)‘𝑦) = ((𝑘𝐴𝑆)‘𝑘))
41, 2, 3cbvixp 7811 . . 3 X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘)
5 simpr 476 . . . . 5 ((𝜑𝑘𝐴) → 𝑘𝐴)
6 elptr2.3 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))
7 eqid 2610 . . . . . 6 (𝑘𝐴𝑆) = (𝑘𝐴𝑆)
87fvmpt2 6200 . . . . 5 ((𝑘𝐴𝑆 ∈ (𝐹𝑘)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
95, 6, 8syl2anc 691 . . . 4 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
109ixpeq2dva 7809 . . 3 (𝜑X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) = X𝑘𝐴 𝑆)
114, 10syl5eq 2656 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 𝑆)
12 elptr2.1 . . 3 (𝜑𝐴𝑉)
136ralrimiva 2949 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ∈ (𝐹𝑘))
147fnmpt 5933 . . . 4 (∀𝑘𝐴 𝑆 ∈ (𝐹𝑘) → (𝑘𝐴𝑆) Fn 𝐴)
1513, 14syl 17 . . 3 (𝜑 → (𝑘𝐴𝑆) Fn 𝐴)
169, 6eqeltrd 2688 . . . . 5 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
1716ralrimiva 2949 . . . 4 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
181nfel1 2765 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦)
19 nfv 1830 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)
20 fveq2 6103 . . . . . 6 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
213, 20eleq12d 2682 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)))
2218, 19, 21cbvral 3143 . . . 4 (∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
2317, 22sylibr 223 . . 3 (𝜑 → ∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦))
24 elptr2.2 . . 3 (𝜑𝑊 ∈ Fin)
25 eldifi 3694 . . . . . . 7 (𝑘 ∈ (𝐴𝑊) → 𝑘𝐴)
2625, 9sylan2 490 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
27 elptr2.4 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))
2826, 27eqtrd 2644 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
2928ralrimiva 2949 . . . 4 (𝜑 → ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
301nfeq1 2764 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦)
31 nfv 1830 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)
3220unieqd 4382 . . . . . 6 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
333, 32eqeq12d 2625 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)))
3430, 31, 33cbvral 3143 . . . 4 (∀𝑦 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
3529, 34sylibr 223 . . 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 21186 . . 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 1327 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ 𝐵)
3911, 38eqeltrrd 2689 1 (𝜑X𝑘𝐴 𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wral 2896  wrex 2897  cdif 3537   cuni 4372  cmpt 4643   Fn wfn 5799  cfv 5804  Xcixp 7794  Fincfn 7841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ixp 7795
This theorem is referenced by:  ptbasid  21188  ptbasin  21190  ptpjpre2  21193  ptopn  21196
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