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Theorem elptr2 21706
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 6422 . . . 4 𝑘((𝑘𝐴𝑆)‘𝑦)
2 nfcv 2941 . . . 4 𝑦((𝑘𝐴𝑆)‘𝑘)
3 fveq2 6411 . . . 4 (𝑦 = 𝑘 → ((𝑘𝐴𝑆)‘𝑦) = ((𝑘𝐴𝑆)‘𝑘))
41, 2, 3cbvixp 8165 . . 3 X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘)
5 simpr 478 . . . . 5 ((𝜑𝑘𝐴) → 𝑘𝐴)
6 elptr2.3 . . . . 5 ((𝜑𝑘𝐴) → 𝑆 ∈ (𝐹𝑘))
7 eqid 2799 . . . . . 6 (𝑘𝐴𝑆) = (𝑘𝐴𝑆)
87fvmpt2 6516 . . . . 5 ((𝑘𝐴𝑆 ∈ (𝐹𝑘)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
95, 6, 8syl2anc 580 . . . 4 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
109ixpeq2dva 8163 . . 3 (𝜑X𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) = X𝑘𝐴 𝑆)
114, 10syl5eq 2845 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) = X𝑘𝐴 𝑆)
12 elptr2.1 . . 3 (𝜑𝐴𝑉)
136ralrimiva 3147 . . . 4 (𝜑 → ∀𝑘𝐴 𝑆 ∈ (𝐹𝑘))
147fnmpt 6231 . . . 4 (∀𝑘𝐴 𝑆 ∈ (𝐹𝑘) → (𝑘𝐴𝑆) Fn 𝐴)
1513, 14syl 17 . . 3 (𝜑 → (𝑘𝐴𝑆) Fn 𝐴)
169, 6eqeltrd 2878 . . . . 5 ((𝜑𝑘𝐴) → ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
1716ralrimiva 3147 . . . 4 (𝜑 → ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
181nfel1 2956 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦)
19 nfv 2010 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)
20 fveq2 6411 . . . . . 6 (𝑦 = 𝑘 → (𝐹𝑦) = (𝐹𝑘))
213, 20eleq12d 2872 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘)))
2218, 19, 21cbvral 3350 . . . 4 (∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦) ↔ ∀𝑘𝐴 ((𝑘𝐴𝑆)‘𝑘) ∈ (𝐹𝑘))
2317, 22sylibr 226 . . 3 (𝜑 → ∀𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ (𝐹𝑦))
24 elptr2.2 . . 3 (𝜑𝑊 ∈ Fin)
25 eldifi 3930 . . . . . . 7 (𝑘 ∈ (𝐴𝑊) → 𝑘𝐴)
2625, 9sylan2 587 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = 𝑆)
27 elptr2.4 . . . . . 6 ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝑆 = (𝐹𝑘))
2826, 27eqtrd 2833 . . . . 5 ((𝜑𝑘 ∈ (𝐴𝑊)) → ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
2928ralrimiva 3147 . . . 4 (𝜑 → ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
301nfeq1 2955 . . . . 5 𝑘((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦)
31 nfv 2010 . . . . 5 𝑦((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)
3220unieqd 4638 . . . . . 6 (𝑦 = 𝑘 (𝐹𝑦) = (𝐹𝑘))
333, 32eqeq12d 2814 . . . . 5 (𝑦 = 𝑘 → (((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘)))
3430, 31, 33cbvral 3350 . . . 4 (∀𝑦 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑦) = (𝐹𝑦) ↔ ∀𝑘 ∈ (𝐴𝑊)((𝑘𝐴𝑆)‘𝑘) = (𝐹𝑘))
3529, 34sylibr 226 . . 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 21705 . . 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 1499 . 2 (𝜑X𝑦𝐴 ((𝑘𝐴𝑆)‘𝑦) ∈ 𝐵)
3911, 38eqeltrrd 2879 1 (𝜑X𝑘𝐴 𝑆𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wex 1875  wcel 2157  {cab 2785  wral 3089  wrex 3090  cdif 3766   cuni 4628  cmpt 4922   Fn wfn 6096  cfv 6101  Xcixp 8148  Fincfn 8195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ixp 8149
This theorem is referenced by:  ptbasid  21707  ptbasin  21709  ptpjpre2  21712  ptopn  21715
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