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Theorem ixpin 6841
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 3358 . . . . . . 7  |-  ( ( f `  x )  e.  ( B  i^i  C )  <->  ( ( f `
 x )  e.  B  /\  ( f `
 x )  e.  C ) )
32ralbii 2567 . . . . . 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 2675 . . . . . 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 2791 . . . . . 6  |-  f  e. 
_V
87elixp 6823 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
97elixp 6823 . . . . 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 6823 . . 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 3358 . . 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 2280 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 1623    e. wcel 1684   A.wral 2543    i^i cin 3151    Fn wfn 5250   ` cfv 5255   X_cixp 6817
This theorem is referenced by:  ptbasin  17272  ptclsg  17309
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263  df-ixp 6818
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