MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elpt Structured version   Visualization version   GIF version

Theorem elpt 23596
Description: Elementhood in the bases of a 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
elpt (𝑆𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑦)))
Distinct variable groups:   𝑔,,𝑤,𝑥,𝑦,𝑧,𝐴   𝑔,𝐹,,𝑤,𝑥,𝑦,𝑧   𝑆,𝑔,,𝑥
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑤,𝑔,)   𝑆(𝑦,𝑧,𝑤)

Proof of Theorem elpt
StepHypRef Expression
1 ptbas.1 . . 3 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
21eleq2i 2831 . 2 (𝑆𝐵𝑆 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))})
3 simpr 484 . . . . 5 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 = X𝑦𝐴 (𝑔𝑦))
4 ixpexg 8961 . . . . . 6 (∀𝑦𝐴 (𝑔𝑦) ∈ V → X𝑦𝐴 (𝑔𝑦) ∈ V)
5 fvexd 6922 . . . . . 6 (𝑦𝐴 → (𝑔𝑦) ∈ V)
64, 5mprg 3065 . . . . 5 X𝑦𝐴 (𝑔𝑦) ∈ V
73, 6eqeltrdi 2847 . . . 4 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
87exlimiv 1928 . . 3 (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
9 eqeq1 2739 . . . . 5 (𝑥 = 𝑆 → (𝑥 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑆 = X𝑦𝐴 (𝑔𝑦)))
109anbi2d 630 . . . 4 (𝑥 = 𝑆 → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦))))
1110exbidv 1919 . . 3 (𝑥 = 𝑆 → (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦))))
128, 11elab3 3689 . 2 (𝑆 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))} ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)))
13 fneq1 6660 . . . . 5 (𝑔 = → (𝑔 Fn 𝐴 Fn 𝐴))
14 fveq1 6906 . . . . . . 7 (𝑔 = → (𝑔𝑦) = (𝑦))
1514eleq1d 2824 . . . . . 6 (𝑔 = → ((𝑔𝑦) ∈ (𝐹𝑦) ↔ (𝑦) ∈ (𝐹𝑦)))
1615ralbidv 3176 . . . . 5 (𝑔 = → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)))
1714eqeq1d 2737 . . . . . . 7 (𝑔 = → ((𝑔𝑦) = (𝐹𝑦) ↔ (𝑦) = (𝐹𝑦)))
1817rexralbidv 3221 . . . . . 6 (𝑔 = → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦)))
19 difeq2 4130 . . . . . . . 8 (𝑧 = 𝑤 → (𝐴𝑧) = (𝐴𝑤))
2019raleqdv 3324 . . . . . . 7 (𝑧 = 𝑤 → (∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)))
2120cbvrexvw 3236 . . . . . 6 (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦))
2218, 21bitrdi 287 . . . . 5 (𝑔 = → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)))
2313, 16, 223anbi123d 1435 . . . 4 (𝑔 = → ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ↔ ( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦))))
2414ixpeq2dv 8952 . . . . 5 (𝑔 = X𝑦𝐴 (𝑔𝑦) = X𝑦𝐴 (𝑦))
2524eqeq2d 2746 . . . 4 (𝑔 = → (𝑆 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑆 = X𝑦𝐴 (𝑦)))
2623, 25anbi12d 632 . . 3 (𝑔 = → (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) ↔ (( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑦))))
2726cbvexvw 2034 . 2 (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑦)))
282, 12, 273bitri 297 1 (𝑆𝐵 ↔ ∃(( Fn 𝐴 ∧ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦) ∧ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  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-rab 3434  df-v 3480  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-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ixp 8937
This theorem is referenced by:  elptr  23597  ptbasin  23601  ptbasfi  23605  ptrecube  37607
  Copyright terms: Public domain W3C validator