Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  inixp Unicode version

Theorem inixp 26123
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 798 . . . 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 626 . . . . 5  |-  ( ( f  Fn  A  /\  f  Fn  A )  <->  f  Fn  A )
3 r19.26 2783 . . . . . 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 3475 . . . . . . . 8  |-  ( ( f `  x )  e.  ( B  i^i  C )  <->  ( ( f `
 x )  e.  B  /\  ( f `
 x )  e.  C ) )
54bicomi 194 . . . . . . 7  |-  ( ( ( f `  x
)  e.  B  /\  ( f `  x
)  e.  C )  <-> 
( f `  x
)  e.  ( B  i^i  C ) )
65ralbii 2675 . . . . . 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 243 . . . . 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 679 . . . 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 241 . . 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 2904 . . . . 5  |-  f  e. 
_V
1110elixp 7007 . . . 4  |-  ( f  e.  X_ x  e.  A  B 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  B ) )
1210elixp 7007 . . . 4  |-  ( f  e.  X_ x  e.  A  C 
<->  ( f  Fn  A  /\  A. x  e.  A  ( f `  x
)  e.  C ) )
1311, 12anbi12i 679 . . 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 7007 . . 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 269 . 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 3479 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 359    = wceq 1649    e. wcel 1717   A.wral 2651    i^i cin 3264    Fn wfn 5391   ` cfv 5396   X_cixp 7001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404  df-ixp 7002
  Copyright terms: Public domain W3C validator