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Theorem inixp 26444
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 799 . . . 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 627 . . . . 5  |-  ( ( f  Fn  A  /\  f  Fn  A )  <->  f  Fn  A )
3 r19.26 2840 . . . . . 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 3532 . . . . . . . 8  |-  ( ( f `  x )  e.  ( B  i^i  C )  <->  ( ( f `
 x )  e.  B  /\  ( f `
 x )  e.  C ) )
54bicomi 195 . . . . . . 7  |-  ( ( ( f `  x
)  e.  B  /\  ( f `  x
)  e.  C )  <-> 
( f `  x
)  e.  ( B  i^i  C ) )
65ralbii 2731 . . . . . 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 244 . . . . 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 680 . . . 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 242 . . 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 2961 . . . . 5  |-  f  e. 
_V
1110elixp 7072 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
1210elixp 7072 . . . 4  |-  ( f  e.  X_ x  e.  A  C 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  C ) )
1311, 12anbi12i 680 . . 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 7072 . . 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 270 . 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 3536 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 set class
Syntax hints:    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    i^i cin 3321    Fn wfn 5452   ` cfv 5457   X_cixp 7066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-iota 5421  df-fun 5459  df-fn 5460  df-fv 5465  df-ixp 7067
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