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Theorem unielxp 6717
Description: The membership relation for a Cartesian product is inherited by union. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
unielxp  |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )

Proof of Theorem unielxp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elxp7 6714 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )
2 elvvuni 5002 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
32adantr 465 . . 3  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  A
)
4 simprl 755 . . . . . 6  |-  ( ( U. A  e.  A  /\  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )  ->  A  e.  ( _V  X.  _V )
)
5 eleq2 2525 . . . . . . . 8  |-  ( x  =  A  ->  ( U. A  e.  x  <->  U. A  e.  A ) )
6 eleq1 2524 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  ( _V 
X.  _V )  <->  A  e.  ( _V  X.  _V )
) )
7 fveq2 5794 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
87eleq1d 2521 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 1st `  x
)  e.  B  <->  ( 1st `  A )  e.  B
) )
9 fveq2 5794 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
109eleq1d 2521 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 2nd `  x
)  e.  C  <->  ( 2nd `  A )  e.  C
) )
118, 10anbi12d 710 . . . . . . . . 9  |-  ( x  =  A  ->  (
( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C )  <->  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) ) )
126, 11anbi12d 710 . . . . . . . 8  |-  ( x  =  A  ->  (
( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) )  <-> 
( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) ) )
135, 12anbi12d 710 . . . . . . 7  |-  ( x  =  A  ->  (
( U. A  e.  x  /\  ( x  e.  ( _V  X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) )  <->  ( U. A  e.  A  /\  ( A  e.  ( _V  X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) ) ) )
1413spcegv 3158 . . . . . 6  |-  ( A  e.  ( _V  X.  _V )  ->  ( ( U. A  e.  A  /\  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )  ->  E. x
( U. A  e.  x  /\  ( x  e.  ( _V  X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) ) ) )
154, 14mpcom 36 . . . . 5  |-  ( ( U. A  e.  A  /\  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )  ->  E. x
( U. A  e.  x  /\  ( x  e.  ( _V  X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) ) )
16 eluniab 4205 . . . . 5  |-  ( U. A  e.  U. { x  |  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x )  e.  B  /\  ( 2nd `  x
)  e.  C ) ) }  <->  E. x
( U. A  e.  x  /\  ( x  e.  ( _V  X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) ) )
1715, 16sylibr 212 . . . 4  |-  ( ( U. A  e.  A  /\  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )  ->  U. A  e. 
U. { x  |  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) } )
18 xp2 6716 . . . . . 6  |-  ( B  X.  C )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  B  /\  ( 2nd `  x
)  e.  C ) }
19 df-rab 2805 . . . . . 6  |-  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  B  /\  ( 2nd `  x )  e.  C
) }  =  {
x  |  ( x  e.  ( _V  X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2018, 19eqtri 2481 . . . . 5  |-  ( B  X.  C )  =  { x  |  ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2120unieqi 4203 . . . 4  |-  U. ( B  X.  C )  = 
U. { x  |  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2217, 21syl6eleqr 2551 . . 3  |-  ( ( U. A  e.  A  /\  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )  ->  U. A  e. 
U. ( B  X.  C ) )
233, 22mpancom 669 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  U. ( B  X.  C
) )
241, 23sylbi 195 1  |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   {cab 2437   {crab 2800   _Vcvv 3072   U.cuni 4194    X. cxp 4941   ` cfv 5521   1stc1st 6680   2ndc2nd 6681
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-iota 5484  df-fun 5523  df-fv 5529  df-1st 6682  df-2nd 6683
This theorem is referenced by: (None)
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