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Theorem ixpin 7079
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 3522 . . . . . . 7  |-  ( ( f `  x )  e.  ( B  i^i  C )  <->  ( ( f `
 x )  e.  B  /\  ( f `
 x )  e.  C ) )
32ralbii 2721 . . . . . 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 2830 . . . . . 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 2951 . . . . . 6  |-  f  e. 
_V
87elixp 7061 . . . . 5  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
97elixp 7061 . . . . 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 7061 . . 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 3522 . . 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 2432 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 1652    e. wcel 1725   A.wral 2697    i^i cin 3311    Fn wfn 5441   ` cfv 5446   X_cixp 7055
This theorem is referenced by:  ptbasin  17601  ptclsg  17639
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454  df-ixp 7056
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