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Theorem elixp 8460
Description: Membership in an infinite Cartesian product. (Contributed by NM, 28-Sep-2006.)
Hypothesis
Ref Expression
elixp.1 𝐹 ∈ V
Assertion
Ref Expression
elixp (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Distinct variable groups:   𝑥,𝐹   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem elixp
StepHypRef Expression
1 elixp2 8457 . 2 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
2 elixp.1 . . 3 𝐹 ∈ V
3 3anass 1089 . . 3 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) ↔ (𝐹 ∈ V ∧ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)))
42, 3mpbiran 705 . 2 ((𝐹 ∈ V ∧ 𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
51, 4bitri 276 1 (𝐹X𝑥𝐴 𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  w3a 1081  wcel 2107  wral 3142  Vcvv 3499   Fn wfn 6346  cfv 6351  Xcixp 8453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fn 6354  df-fv 6359  df-ixp 8454
This theorem is referenced by:  elixpconst  8461  ixpin  8479  ixpiin  8480  resixpfo  8492  elixpsn  8493  boxriin  8496  boxcutc  8497  ixpfi2  8814  ixpiunwdom  9047  dfac9  9554  ac9  9897  ac9s  9907  konigthlem  9982  cofucl  17150  yonedalem3  17522  psrbaglefi  20073  ptpjpre1  22097  ptpjcn  22137  ptpjopn  22138  ptclsg  22141  dfac14  22144  pthaus  22164  xkopt  22181  ptcmplem2  22579  ptcmplem3  22580  ptcmplem4  22581  prdsbl  23018  prdsxmslem2  23056  eulerpartlemb  31514  ptpconn  32366  finixpnum  34746  ptrest  34760  poimirlem29  34790  poimirlem30  34791  inixp  34873  prdstotbnd  34942  ioorrnopnlem  42457  hoicvr  42698  hoidmvlelem3  42747  hspdifhsp  42766  hspmbllem2  42777
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