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Theorem inixp 28634
Description: Intersection of Cartesian products over the same base set. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
inixp  |-  ( X_ x  e.  A  B  i^i  X_ x  e.  A  C )  =  X_ x  e.  A  ( B  i^i  C )
Distinct variable group:    x, A
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem inixp
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 an4 820 . . . 4  |-  ( ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B )  /\  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  C
) )  <->  ( (
f  Fn  A  /\  f  Fn  A )  /\  ( A. x  e.  A  ( f `  x )  e.  B  /\  A. x  e.  A  ( f `  x
)  e.  C ) ) )
2 anidm 644 . . . . 5  |-  ( ( f  Fn  A  /\  f  Fn  A )  <->  f  Fn  A )
3 r19.26 2861 . . . . . 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 ) )
4 elin 3551 . . . . . . . 8  |-  ( ( f `  x )  e.  ( B  i^i  C )  <->  ( ( f `
 x )  e.  B  /\  ( f `
 x )  e.  C ) )
54bicomi 202 . . . . . . 7  |-  ( ( ( f `  x
)  e.  B  /\  ( f `  x
)  e.  C )  <-> 
( f `  x
)  e.  ( B  i^i  C ) )
65ralbii 2751 . . . . . 6  |-  ( A. x  e.  A  (
( f `  x
)  e.  B  /\  ( f `  x
)  e.  C )  <->  A. x  e.  A  ( f `  x
)  e.  ( B  i^i  C ) )
73, 6bitr3i 251 . . . . 5  |-  ( ( A. x  e.  A  ( f `  x
)  e.  B  /\  A. x  e.  A  ( f `  x )  e.  C )  <->  A. x  e.  A  ( f `  x )  e.  ( B  i^i  C ) )
82, 7anbi12i 697 . . . 4  |-  ( ( ( f  Fn  A  /\  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  i^i  C ) ) )
91, 8bitri 249 . . 3  |-  ( ( ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B )  /\  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  C
) )  <->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x )  e.  ( B  i^i  C ) ) )
10 vex 2987 . . . . 5  |-  f  e. 
_V
1110elixp 7282 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
1210elixp 7282 . . . 4  |-  ( f  e.  X_ x  e.  A  C 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  C ) )
1311, 12anbi12i 697 . . 3  |-  ( ( 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 ) ) )
1410elixp 7282 . . 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 ) ) )
159, 13, 143bitr4i 277 . 2  |-  ( ( f  e.  X_ x  e.  A  B  /\  f  e.  X_ x  e.  A  C )  <->  f  e.  X_ x  e.  A  ( B  i^i  C ) )
1615ineqri 3556 1  |-  ( X_ x  e.  A  B  i^i  X_ x  e.  A  C )  =  X_ x  e.  A  ( B  i^i  C )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2727    i^i cin 3339    Fn wfn 5425   ` cfv 5430   X_cixp 7275
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5393  df-fun 5432  df-fn 5433  df-fv 5438  df-ixp 7276
This theorem is referenced by: (None)
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