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Theorem elpt 21369
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 2692 . 2 (𝑆𝐵𝑆 ∈ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))})
3 simpr 477 . . . . 5 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 = X𝑦𝐴 (𝑔𝑦))
4 ixpexg 7929 . . . . . 6 (∀𝑦𝐴 (𝑔𝑦) ∈ V → X𝑦𝐴 (𝑔𝑦) ∈ V)
5 fvexd 6201 . . . . . 6 (𝑦𝐴 → (𝑔𝑦) ∈ V)
64, 5mprg 2925 . . . . 5 X𝑦𝐴 (𝑔𝑦) ∈ V
73, 6syl6eqel 2708 . . . 4 (((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
87exlimiv 1857 . . 3 (∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑆 = X𝑦𝐴 (𝑔𝑦)) → 𝑆 ∈ V)
9 eqeq1 2625 . . . . 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 1849 . . 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 3356 . 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 5977 . . . . 5 (𝑔 = → (𝑔 Fn 𝐴 Fn 𝐴))
14 fveq1 6188 . . . . . . 7 (𝑔 = → (𝑔𝑦) = (𝑦))
1514eleq1d 2685 . . . . . 6 (𝑔 = → ((𝑔𝑦) ∈ (𝐹𝑦) ↔ (𝑦) ∈ (𝐹𝑦)))
1615ralbidv 2985 . . . . 5 (𝑔 = → (∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ↔ ∀𝑦𝐴 (𝑦) ∈ (𝐹𝑦)))
1714eqeq1d 2623 . . . . . . 7 (𝑔 = → ((𝑔𝑦) = (𝐹𝑦) ↔ (𝑦) = (𝐹𝑦)))
1817rexralbidv 3056 . . . . . 6 (𝑔 = → (∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦)))
19 difeq2 3720 . . . . . . . 8 (𝑧 = 𝑤 → (𝐴𝑧) = (𝐴𝑤))
2019raleqdv 3142 . . . . . . 7 (𝑧 = 𝑤 → (∀𝑦 ∈ (𝐴𝑧)(𝑦) = (𝐹𝑦) ↔ ∀𝑦 ∈ (𝐴𝑤)(𝑦) = (𝐹𝑦)))
2120cbvrexv 3170 . . . . . 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 1398 . . . 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 7921 . . . . 5 (𝑔 = X𝑦𝐴 (𝑔𝑦) = X𝑦𝐴 (𝑦))
2524eqeq2d 2631 . . . 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 2274 . 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 384  w3a 1037   = wceq 1482  wex 1703  wcel 1989  {cab 2607  wral 2911  wrex 2912  Vcvv 3198  cdif 3569   cuni 4434   Fn wfn 5881  cfv 5886  Xcixp 7905  Fincfn 7952
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ixp 7906
This theorem is referenced by:  elptr  21370  ptbasin  21374  ptbasfi  21378  ptrecube  33389
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