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Theorem elpt 23601
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 2836 . 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 8980 . . . . . 6 (∀𝑦𝐴 (𝑔𝑦) ∈ V → X𝑦𝐴 (𝑔𝑦) ∈ V)
5 fvexd 6935 . . . . . 6 (𝑦𝐴 → (𝑔𝑦) ∈ V)
64, 5mprg 3073 . . . . 5 X𝑦𝐴 (𝑔𝑦) ∈ V
73, 6eqeltrdi 2852 . . . 4 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
87exlimiv 1929 . . 3 (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
9 eqeq1 2744 . . . . 5 (𝑥 = 𝑆 → (𝑥 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑆 = X𝑦𝐴 (𝑔𝑦)))
109anbi2d 629 . . . 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 1920 . . 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 3702 . 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 6670 . . . . 5 (𝑔 = → (𝑔 Fn 𝐴 Fn 𝐴))
14 fveq1 6919 . . . . . . 7 (𝑔 = → (𝑔𝑦) = (𝑦))
1514eleq1d 2829 . . . . . 6 (𝑔 = → ((𝑔𝑦) ∈ (𝐹𝑦) ↔ (𝑦) ∈ (𝐹𝑦)))
1615ralbidv 3184 . . . . 5 (𝑔 = → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)))
1714eqeq1d 2742 . . . . . . 7 (𝑔 = → ((𝑔𝑦) = (𝐹𝑦) ↔ (𝑦) = (𝐹𝑦)))
1817rexralbidv 3229 . . . . . 6 (𝑔 = → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦)))
19 difeq2 4143 . . . . . . . 8 (𝑧 = 𝑤 → (𝐴𝑧) = (𝐴𝑤))
2019raleqdv 3334 . . . . . . 7 (𝑧 = 𝑤 → (∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)))
2120cbvrexvw 3244 . . . . . 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 1436 . . . 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 8971 . . . . 5 (𝑔 = X𝑦𝐴 (𝑔𝑦) = X𝑦𝐴 (𝑦))
2524eqeq2d 2751 . . . 4 (𝑔 = → (𝑆 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑆 = X𝑦𝐴 (𝑦)))
2623, 25anbi12d 631 . . 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 2036 . 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 1087   = wceq 1537  wex 1777  wcel 2108  {cab 2717  wral 3067  wrex 3076  Vcvv 3488  cdif 3973   cuni 4931   Fn wfn 6568  cfv 6573  Xcixp 8955  Fincfn 9003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ixp 8956
This theorem is referenced by:  elptr  23602  ptbasin  23606  ptbasfi  23610  ptrecube  37580
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