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Theorem elpt 21423
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 2722 . 2 (𝑆𝐵𝑆 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))})
3 simpr 476 . . . . 5 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 = X𝑦𝐴 (𝑔𝑦))
4 ixpexg 7974 . . . . . 6 (∀𝑦𝐴 (𝑔𝑦) ∈ V → X𝑦𝐴 (𝑔𝑦) ∈ V)
5 fvexd 6241 . . . . . 6 (𝑦𝐴 → (𝑔𝑦) ∈ V)
64, 5mprg 2955 . . . . 5 X𝑦𝐴 (𝑔𝑦) ∈ V
73, 6syl6eqel 2738 . . . 4 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
87exlimiv 1898 . . 3 (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
9 eqeq1 2655 . . . . 5 (𝑥 = 𝑆 → (𝑥 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑆 = X𝑦𝐴 (𝑔𝑦)))
109anbi2d 740 . . . 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 1890 . . 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 3390 . 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 6017 . . . . 5 (𝑔 = → (𝑔 Fn 𝐴 Fn 𝐴))
14 fveq1 6228 . . . . . . 7 (𝑔 = → (𝑔𝑦) = (𝑦))
1514eleq1d 2715 . . . . . 6 (𝑔 = → ((𝑔𝑦) ∈ (𝐹𝑦) ↔ (𝑦) ∈ (𝐹𝑦)))
1615ralbidv 3015 . . . . 5 (𝑔 = → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)))
1714eqeq1d 2653 . . . . . . 7 (𝑔 = → ((𝑔𝑦) = (𝐹𝑦) ↔ (𝑦) = (𝐹𝑦)))
1817rexralbidv 3087 . . . . . 6 (𝑔 = → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦)))
19 difeq2 3755 . . . . . . . 8 (𝑧 = 𝑤 → (𝐴𝑧) = (𝐴𝑤))
2019raleqdv 3174 . . . . . . 7 (𝑧 = 𝑤 → (∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)))
2120cbvrexv 3202 . . . . . 6 (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦))
2218, 21syl6bb 276 . . . . 5 (𝑔 = → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)))
2313, 16, 223anbi123d 1439 . . . 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 7966 . . . . 5 (𝑔 = X𝑦𝐴 (𝑔𝑦) = X𝑦𝐴 (𝑦))
2524eqeq2d 2661 . . . 4 (𝑔 = → (𝑆 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑆 = X𝑦𝐴 (𝑦)))
2623, 25anbi12d 747 . . 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))))
2726cbvexv 2311 . 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 286 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 196  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  cdif 3604   cuni 4468   Fn wfn 5921  cfv 5926  Xcixp 7950  Fincfn 7997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ixp 7951
This theorem is referenced by:  elptr  21424  ptbasin  21428  ptbasfi  21432  ptrecube  33539
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