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Theorem elptr 23421
Description: A basic open set in the product topology. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypothesis
Ref Expression
ptbas.1 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
Assertion
Ref Expression
elptr ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝐺𝑦) ∈ 𝐵)
Distinct variable groups:   𝑥,𝑔,𝑦,𝐺   𝑧,𝑔,𝐴,𝑥,𝑦   𝑔,𝐹,𝑥,𝑦,𝑧   𝑔,𝑉,𝑥,𝑦,𝑧   𝑦,𝑊
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑔)   𝐺(𝑧)   𝑊(𝑥,𝑧,𝑔)

Proof of Theorem elptr
Dummy variables 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2l 1196 . . . 4 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → 𝐺 Fn 𝐴)
2 simp1 1133 . . . 4 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → 𝐴𝑉)
31, 2fnexd 7212 . . 3 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → 𝐺 ∈ V)
4 simp2r 1197 . . . 4 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦))
5 difeq2 4109 . . . . . . 7 (𝑤 = 𝑊 → (𝐴𝑤) = (𝐴𝑊))
65raleqdv 3317 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦)))
76rspcev 3604 . . . . 5 ((𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦)) → ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦))
873ad2ant3 1132 . . . 4 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦))
91, 4, 83jca 1125 . . 3 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦)))
10 fveq1 6881 . . . . . . 7 ( = 𝐺 → (𝑦) = (𝐺𝑦))
1110eqcomd 2730 . . . . . 6 ( = 𝐺 → (𝐺𝑦) = (𝑦))
1211ixpeq2dv 8904 . . . . 5 ( = 𝐺X𝑦𝐴 (𝐺𝑦) = X𝑦𝐴 (𝑦))
1312biantrud 531 . . . 4 ( = 𝐺 → (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ↔ (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ X𝑦𝐴 (𝐺𝑦) = X𝑦𝐴 (𝑦))))
14 fneq1 6631 . . . . 5 ( = 𝐺 → ( Fn 𝐴𝐺 Fn 𝐴))
1510eleq1d 2810 . . . . . 6 ( = 𝐺 → ((𝑦) ∈ (𝐹𝑦) ↔ (𝐺𝑦) ∈ (𝐹𝑦)))
1615ralbidv 3169 . . . . 5 ( = 𝐺 → (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ↔ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)))
1710eqeq1d 2726 . . . . . 6 ( = 𝐺 → ((𝑦) = (𝐹𝑦) ↔ (𝐺𝑦) = (𝐹𝑦)))
1817rexralbidv 3212 . . . . 5 ( = 𝐺 → (∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦)))
1914, 16, 183anbi123d 1432 . . . 4 ( = 𝐺 → (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ↔ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦))))
2013, 19bitr3d 281 . . 3 ( = 𝐺 → ((( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ X𝑦𝐴 (𝐺𝑦) = X𝑦𝐴 (𝑦)) ↔ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦))))
213, 9, 20spcedv 3580 . 2 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ X𝑦𝐴 (𝐺𝑦) = X𝑦𝐴 (𝑦)))
22 ptbas.1 . . 3 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
2322elpt 23420 . 2 (X𝑦𝐴 (𝐺𝑦) ∈ 𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ X𝑦𝐴 (𝐺𝑦) = X𝑦𝐴 (𝑦)))
2421, 23sylibr 233 1 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → X𝑦𝐴 (𝐺𝑦) ∈ 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2701  wral 3053  wrex 3062  Vcvv 3466  cdif 3938   cuni 4900   Fn wfn 6529  cfv 6534  Xcixp 8888  Fincfn 8936
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ixp 8889
This theorem is referenced by:  elptr2  23422
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