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Theorem inixp 32119
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 840 . . . 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 656 . . . . 5  |-  ( ( f  Fn  A  /\  f  Fn  A )  <->  f  Fn  A )
3 r19.26 2904 . . . . . 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 3608 . . . . . . . 8  |-  ( ( f `  x )  e.  ( B  i^i  C )  <->  ( ( f `
 x )  e.  B  /\  ( f `
 x )  e.  C ) )
54bicomi 207 . . . . . . 7  |-  ( ( ( f `  x
)  e.  B  /\  ( f `  x
)  e.  C )  <-> 
( f `  x
)  e.  ( B  i^i  C ) )
65ralbii 2823 . . . . . 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 259 . . . . 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 711 . . . 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 257 . . 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 3034 . . . . 5  |-  f  e. 
_V
1110elixp 7547 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
1210elixp 7547 . . . 4  |-  ( f  e.  X_ x  e.  A  C 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  C ) )
1311, 12anbi12i 711 . . 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 7547 . . 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 285 . 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 3617 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 376    = wceq 1452    e. wcel 1904   A.wral 2756    i^i cin 3389    Fn wfn 5584   ` cfv 5589   X_cixp 7540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fn 5592  df-fv 5597  df-ixp 7541
This theorem is referenced by: (None)
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