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Theorem elptr 23597
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 1198 . . . 4 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → 𝐺 Fn 𝐴)
2 simp1 1135 . . . 4 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → 𝐴𝑉)
31, 2fnexd 7238 . . 3 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → 𝐺 ∈ V)
4 simp2r 1199 . . . 4 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦))
5 difeq2 4130 . . . . . . 7 (𝑤 = 𝑊 → (𝐴𝑤) = (𝐴𝑊))
65raleqdv 3324 . . . . . 6 (𝑤 = 𝑊 → (∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦)))
76rspcev 3622 . . . . 5 ((𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦)) → ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦))
873ad2ant3 1134 . . . 4 ((𝐴𝑉 ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)) ∧ (𝑊 ∈ Fin ∧ ∀𝑦 ∈ (𝐴𝑊)(𝐺𝑦) = (𝐹𝑦))) → ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦))
91, 4, 83jca 1127 . . 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 6906 . . . . . . 7 ( = 𝐺 → (𝑦) = (𝐺𝑦))
1110eqcomd 2741 . . . . . 6 ( = 𝐺 → (𝐺𝑦) = (𝑦))
1211ixpeq2dv 8952 . . . . 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 6660 . . . . 5 ( = 𝐺 → ( Fn 𝐴𝐺 Fn 𝐴))
1510eleq1d 2824 . . . . . 6 ( = 𝐺 → ((𝑦) ∈ (𝐹𝑦) ↔ (𝐺𝑦) ∈ (𝐹𝑦)))
1615ralbidv 3176 . . . . 5 ( = 𝐺 → (∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ↔ ∀𝑦𝐴 (𝐺𝑦) ∈ (𝐹𝑦)))
1710eqeq1d 2737 . . . . . 6 ( = 𝐺 → ((𝑦) = (𝐹𝑦) ↔ (𝐺𝑦) = (𝐹𝑦)))
1817rexralbidv 3221 . . . . 5 ( = 𝐺 → (∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝐺𝑦) = (𝐹𝑦)))
1914, 16, 183anbi123d 1435 . . . 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 3598 . 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 23596 . 2 (X𝑦𝐴 (𝐺𝑦) ∈ 𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ X𝑦𝐴 (𝐺𝑦) = X𝑦𝐴 (𝑦)))
2421, 23sylibr 234 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 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wral 3059  wrex 3068  Vcvv 3478  cdif 3960   cuni 4912   Fn wfn 6558  cfv 6563  Xcixp 8936  Fincfn 8984
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ixp 8937
This theorem is referenced by:  elptr2  23598
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