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Theorem elpt 22191
 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 2881 . 2 (𝑆𝐵𝑆 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))})
3 simpr 488 . . . . 5 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 = X𝑦𝐴 (𝑔𝑦))
4 ixpexg 8476 . . . . . 6 (∀𝑦𝐴 (𝑔𝑦) ∈ V → X𝑦𝐴 (𝑔𝑦) ∈ V)
5 fvexd 6665 . . . . . 6 (𝑦𝐴 → (𝑔𝑦) ∈ V)
64, 5mprg 3120 . . . . 5 X𝑦𝐴 (𝑔𝑦) ∈ V
73, 6eqeltrdi 2898 . . . 4 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
87exlimiv 1931 . . 3 (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
9 eqeq1 2802 . . . . 5 (𝑥 = 𝑆 → (𝑥 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑆 = X𝑦𝐴 (𝑔𝑦)))
109anbi2d 631 . . . 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 1922 . . 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 3622 . 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 6417 . . . . 5 (𝑔 = → (𝑔 Fn 𝐴 Fn 𝐴))
14 fveq1 6649 . . . . . . 7 (𝑔 = → (𝑔𝑦) = (𝑦))
1514eleq1d 2874 . . . . . 6 (𝑔 = → ((𝑔𝑦) ∈ (𝐹𝑦) ↔ (𝑦) ∈ (𝐹𝑦)))
1615ralbidv 3162 . . . . 5 (𝑔 = → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)))
1714eqeq1d 2800 . . . . . . 7 (𝑔 = → ((𝑔𝑦) = (𝐹𝑦) ↔ (𝑦) = (𝐹𝑦)))
1817rexralbidv 3260 . . . . . 6 (𝑔 = → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦)))
19 difeq2 4044 . . . . . . . 8 (𝑧 = 𝑤 → (𝐴𝑧) = (𝐴𝑤))
2019raleqdv 3364 . . . . . . 7 (𝑧 = 𝑤 → (∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)))
2120cbvrexvw 3397 . . . . . 6 (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦))
2218, 21syl6bb 290 . . . . 5 (𝑔 = → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦) ↔ ∃𝑤 ∈ Fin ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)))
2313, 16, 223anbi123d 1433 . . . 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 8467 . . . . 5 (𝑔 = X𝑦𝐴 (𝑔𝑦) = X𝑦𝐴 (𝑦))
2524eqeq2d 2809 . . . 4 (𝑔 = → (𝑆 = X𝑦𝐴 (𝑔𝑦) ↔ 𝑆 = X𝑦𝐴 (𝑦)))
2623, 25anbi12d 633 . . 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 2044 . 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 300 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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∖ cdif 3878  ∪ cuni 4801   Fn wfn 6322  ‘cfv 6327  Xcixp 8451  Fincfn 8499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ixp 8452 This theorem is referenced by:  elptr  22192  ptbasin  22196  ptbasfi  22200  ptrecube  35097
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