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Theorem unielxp 6168
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 6164 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )
2 elvvuni 4686 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
32adantr 276 . . 3  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  A
)
4 simprl 529 . . . . . 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 2241 . . . . . . . 8  |-  ( x  =  A  ->  ( U. A  e.  x  <->  U. A  e.  A ) )
6 eleq1 2240 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  ( _V 
X.  _V )  <->  A  e.  ( _V  X.  _V )
) )
7 fveq2 5510 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
87eleq1d 2246 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 1st `  x
)  e.  B  <->  ( 1st `  A )  e.  B
) )
9 fveq2 5510 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
109eleq1d 2246 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 2nd `  x
)  e.  C  <->  ( 2nd `  A )  e.  C
) )
118, 10anbi12d 473 . . . . . . . . 9  |-  ( x  =  A  ->  (
( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C )  <->  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) ) )
126, 11anbi12d 473 . . . . . . . 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 473 . . . . . . 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 2825 . . . . . 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 3819 . . . . 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 134 . . . 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 6167 . . . . . 6  |-  ( B  X.  C )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  B  /\  ( 2nd `  x
)  e.  C ) }
19 df-rab 2464 . . . . . 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 2198 . . . . 5  |-  ( B  X.  C )  =  { x  |  ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2120unieqi 3817 . . . 4  |-  U. ( B  X.  C )  = 
U. { x  |  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2217, 21eleqtrrdi 2271 . . 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 422 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  U. ( B  X.  C
) )
241, 23sylbi 121 1  |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353   E.wex 1492    e. wcel 2148   {cab 2163   {crab 2459   _Vcvv 2737   U.cuni 3807    X. cxp 4620   ` cfv 5211   1stc1st 6132   2ndc2nd 6133
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fo 5217  df-fv 5219  df-1st 6134  df-2nd 6135
This theorem is referenced by: (None)
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