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Theorem unielxp 6388
Description: The membership relation for a cross 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 6382 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )
2 elvvuni 4941 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
32adantr 453 . . 3  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  A
)
4 simprl 734 . . . . . 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 2499 . . . . . . . 8  |-  ( x  =  A  ->  ( U. A  e.  x  <->  U. A  e.  A ) )
6 eleq1 2498 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  ( _V 
X.  _V )  <->  A  e.  ( _V  X.  _V )
) )
7 fveq2 5731 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
87eleq1d 2504 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 1st `  x
)  e.  B  <->  ( 1st `  A )  e.  B
) )
9 fveq2 5731 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
109eleq1d 2504 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 2nd `  x
)  e.  C  <->  ( 2nd `  A )  e.  C
) )
118, 10anbi12d 693 . . . . . . . . 9  |-  ( x  =  A  ->  (
( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C )  <->  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) ) )
126, 11anbi12d 693 . . . . . . . 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 693 . . . . . . 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 3039 . . . . . 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 35 . . . . 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 4029 . . . . 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 205 . . . 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 6387 . . . . . 6  |-  ( B  X.  C )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  B  /\  ( 2nd `  x
)  e.  C ) }
19 df-rab 2716 . . . . . 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 2458 . . . . 5  |-  ( B  X.  C )  =  { x  |  ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2120unieqi 4027 . . . 4  |-  U. ( B  X.  C )  = 
U. { x  |  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2217, 21syl6eleqr 2529 . . 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 652 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  U. ( B  X.  C
) )
241, 23sylbi 189 1  |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   {crab 2711   _Vcvv 2958   U.cuni 4017    X. cxp 4879   ` cfv 5457   1stc1st 6350   2ndc2nd 6351
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-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
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-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  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-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-iota 5421  df-fun 5459  df-fv 5465  df-1st 6352  df-2nd 6353
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