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Theorem unielxp 6242
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 6236 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )
2 elvvuni 4829 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
32adantr 451 . . 3  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  A
)
4 simprl 732 . . . . . 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 2419 . . . . . . . 8  |-  ( x  =  A  ->  ( U. A  e.  x  <->  U. A  e.  A ) )
6 eleq1 2418 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  ( _V 
X.  _V )  <->  A  e.  ( _V  X.  _V )
) )
7 fveq2 5605 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
87eleq1d 2424 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 1st `  x
)  e.  B  <->  ( 1st `  A )  e.  B
) )
9 fveq2 5605 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
109eleq1d 2424 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 2nd `  x
)  e.  C  <->  ( 2nd `  A )  e.  C
) )
118, 10anbi12d 691 . . . . . . . . 9  |-  ( x  =  A  ->  (
( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C )  <->  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) ) )
126, 11anbi12d 691 . . . . . . . 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 691 . . . . . . 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 2945 . . . . . 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 32 . . . . 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 3918 . . . . 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 203 . . . 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 6241 . . . . . 6  |-  ( B  X.  C )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  B  /\  ( 2nd `  x
)  e.  C ) }
19 df-rab 2628 . . . . . 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 2378 . . . . 5  |-  ( B  X.  C )  =  { x  |  ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2120unieqi 3916 . . . 4  |-  U. ( B  X.  C )  = 
U. { x  |  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2217, 21syl6eleqr 2449 . . 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 650 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  U. ( B  X.  C
) )
241, 23sylbi 187 1  |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   E.wex 1541    = wceq 1642    e. wcel 1710   {cab 2344   {crab 2623   _Vcvv 2864   U.cuni 3906    X. cxp 4766   ` cfv 5334   1stc1st 6204   2ndc2nd 6205
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-mpt 4158  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fv 5342  df-1st 6206  df-2nd 6207
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