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Theorem elptr2 22098
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 6678 . . . 4 𝑘((𝑘𝐴𝑆)‘𝑦)
2 nfcv 2982 . . . 4 𝑦((𝑘𝐴𝑆)‘𝑘)
3 fveq2 6667 . . . 4 (𝑦 = 𝑘 → ((𝑘𝐴𝑆)‘𝑦) = ((𝑘𝐴𝑆)‘𝑘))
41, 2, 3cbvixp 8467 . . 3 X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘)
5 simpr 485 . . . . 5 ((𝜑𝑘𝐴) → 𝑘𝐴)
6 elptr2.3 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))
7 eqid 2826 . . . . . 6 (𝑘𝐴𝑆) = (𝑘𝐴𝑆)
87fvmpt2 6775 . . . . 5 ((𝑘𝐴𝑆 ∈ (𝐹𝑘)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
95, 6, 8syl2anc 584 . . . 4 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
109ixpeq2dva 8465 . . 3 (𝜑X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) = X𝑘𝐴 𝑆)
114, 10syl5eq 2873 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 𝑆)
12 elptr2.1 . . 3 (𝜑𝐴𝑉)
136ralrimiva 3187 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ∈ (𝐹𝑘))
147fnmpt 6485 . . . 4 (∀𝑘𝐴 𝑆 ∈ (𝐹𝑘) → (𝑘𝐴𝑆) Fn 𝐴)
1513, 14syl 17 . . 3 (𝜑 → (𝑘𝐴𝑆) Fn 𝐴)
169, 6eqeltrd 2918 . . . . 5 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
1716ralrimiva 3187 . . . 4 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
181nfel1 2999 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦)
19 nfv 1908 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)
20 fveq2 6667 . . . . . 6 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
213, 20eleq12d 2912 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)))
2218, 19, 21cbvralw 3447 . . . 4 (∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
2317, 22sylibr 235 . . 3 (𝜑 → ∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦))
24 elptr2.2 . . 3 (𝜑𝑊 ∈ Fin)
25 eldifi 4107 . . . . . . 7 (𝑘 ∈ (𝐴𝑊) → 𝑘𝐴)
2625, 9sylan2 592 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
27 elptr2.4 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))
2826, 27eqtrd 2861 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
2928ralrimiva 3187 . . . 4 (𝜑 → ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
301nfeq1 2998 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦)
31 nfv 1908 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)
3220unieqd 4847 . . . . . 6 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
333, 32eqeq12d 2842 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)))
3430, 31, 33cbvralw 3447 . . . 4 (∀𝑦 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
3529, 34sylibr 235 . . 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 22097 . . 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 1373 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ 𝐵)
3911, 38eqeltrrd 2919 1 (𝜑X𝑘𝐴 𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  {cab 2804  wral 3143  wrex 3144  cdif 3937   cuni 4837  cmpt 5143   Fn wfn 6347  cfv 6352  Xcixp 8450  Fincfn 8498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-ixp 8451
This theorem is referenced by:  ptbasid  22099  ptbasin  22101  ptpjpre2  22104  ptopn  22107
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