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Theorem ixpin 7023
Description: The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)
Assertion
Ref Expression
ixpin  |-  X_ x  e.  A  ( B  i^i  C )  =  (
X_ x  e.  A  B  i^i  X_ x  e.  A  C )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem ixpin
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 anandi 802 . . . 4  |-  ( ( f  Fn  A  /\  ( A. x  e.  A  ( f `  x
)  e.  B  /\  A. x  e.  A  ( f `  x )  e.  C ) )  <-> 
( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B
)  /\  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  C
) ) )
2 elin 3473 . . . . . . 7  |-  ( ( f `  x )  e.  ( B  i^i  C )  <->  ( ( f `
 x )  e.  B  /\  ( f `
 x )  e.  C ) )
32ralbii 2673 . . . . . 6  |-  ( A. x  e.  A  (
f `  x )  e.  ( B  i^i  C
)  <->  A. x  e.  A  ( ( f `  x )  e.  B  /\  ( f `  x
)  e.  C ) )
4 r19.26 2781 . . . . . 6  |-  ( A. x  e.  A  (
( f `  x
)  e.  B  /\  ( f `  x
)  e.  C )  <-> 
( A. x  e.  A  ( f `  x )  e.  B  /\  A. x  e.  A  ( f `  x
)  e.  C ) )
53, 4bitri 241 . . . . 5  |-  ( A. x  e.  A  (
f `  x )  e.  ( B  i^i  C
)  <->  ( A. x  e.  A  ( f `  x )  e.  B  /\  A. x  e.  A  ( f `  x
)  e.  C ) )
65anbi2i 676 . . . 4  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  ( B  i^i  C ) )  <->  ( f  Fn  A  /\  ( A. x  e.  A  ( f `  x
)  e.  B  /\  A. x  e.  A  ( f `  x )  e.  C ) ) )
7 vex 2902 . . . . . 6  |-  f  e. 
_V
87elixp 7005 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
97elixp 7005 . . . . 5  |-  ( f  e.  X_ x  e.  A  C 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  C ) )
108, 9anbi12i 679 . . . 4  |-  ( ( f  e.  X_ x  e.  A  B  /\  f  e.  X_ x  e.  A  C )  <->  ( (
f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  B )  /\  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  C ) ) )
111, 6, 103bitr4i 269 . . 3  |-  ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  ( B  i^i  C ) )  <->  ( f  e.  X_ x  e.  A  B  /\  f  e.  X_ x  e.  A  C
) )
127elixp 7005 . . 3  |-  ( f  e.  X_ x  e.  A  ( B  i^i  C )  <-> 
( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  ( B  i^i  C ) ) )
13 elin 3473 . . 3  |-  ( f  e.  ( X_ x  e.  A  B  i^i  X_ x  e.  A  C
)  <->  ( f  e.  X_ x  e.  A  B  /\  f  e.  X_ x  e.  A  C
) )
1411, 12, 133bitr4i 269 . 2  |-  ( f  e.  X_ x  e.  A  ( B  i^i  C )  <-> 
f  e.  ( X_ x  e.  A  B  i^i  X_ x  e.  A  C ) )
1514eqriv 2384 1  |-  X_ x  e.  A  ( B  i^i  C )  =  (
X_ x  e.  A  B  i^i  X_ x  e.  A  C )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649    i^i cin 3262    Fn wfn 5389   ` cfv 5394   X_cixp 6999
This theorem is referenced by:  ptbasin  17530  ptclsg  17568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fn 5397  df-fv 5402  df-ixp 7000
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