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Theorem ixpin 6857
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 801 . . . 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 3371 . . . . . . 7  |-  ( ( f `  x )  e.  ( B  i^i  C )  <->  ( ( f `
 x )  e.  B  /\  ( f `
 x )  e.  C ) )
32ralbii 2580 . . . . . 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 2688 . . . . . 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 240 . . . . 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 675 . . . 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 2804 . . . . . 6  |-  f  e. 
_V
87elixp 6839 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
97elixp 6839 . . . . 5  |-  ( f  e.  X_ x  e.  A  C 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  C ) )
108, 9anbi12i 678 . . . 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 268 . . 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 6839 . . 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 3371 . . 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 268 . 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 2293 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 358    = wceq 1632    e. wcel 1696   A.wral 2556    i^i cin 3164    Fn wfn 5266   ` cfv 5271   X_cixp 6833
This theorem is referenced by:  ptbasin  17288  ptclsg  17325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-ixp 6834
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