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Theorem unielxp 6079
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 6075 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )
2 elvvuni 4610 . . . 4  |-  ( A  e.  ( _V  X.  _V )  ->  U. A  e.  A )
32adantr 274 . . 3  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  A
)
4 simprl 521 . . . . . 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 2204 . . . . . . . 8  |-  ( x  =  A  ->  ( U. A  e.  x  <->  U. A  e.  A ) )
6 eleq1 2203 . . . . . . . . 9  |-  ( x  =  A  ->  (
x  e.  ( _V 
X.  _V )  <->  A  e.  ( _V  X.  _V )
) )
7 fveq2 5428 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 1st `  x )  =  ( 1st `  A
) )
87eleq1d 2209 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 1st `  x
)  e.  B  <->  ( 1st `  A )  e.  B
) )
9 fveq2 5428 . . . . . . . . . . 11  |-  ( x  =  A  ->  ( 2nd `  x )  =  ( 2nd `  A
) )
109eleq1d 2209 . . . . . . . . . 10  |-  ( x  =  A  ->  (
( 2nd `  x
)  e.  C  <->  ( 2nd `  A )  e.  C
) )
118, 10anbi12d 465 . . . . . . . . 9  |-  ( x  =  A  ->  (
( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C )  <->  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) ) )
126, 11anbi12d 465 . . . . . . . 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 465 . . . . . . 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 2777 . . . . . 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 3755 . . . . 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 133 . . . 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 6078 . . . . . 6  |-  ( B  X.  C )  =  { x  e.  ( _V  X.  _V )  |  ( ( 1st `  x )  e.  B  /\  ( 2nd `  x
)  e.  C ) }
19 df-rab 2426 . . . . . 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 2161 . . . . 5  |-  ( B  X.  C )  =  { x  |  ( x  e.  ( _V 
X.  _V )  /\  (
( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2120unieqi 3753 . . . 4  |-  U. ( B  X.  C )  = 
U. { x  |  ( x  e.  ( _V  X.  _V )  /\  ( ( 1st `  x
)  e.  B  /\  ( 2nd `  x )  e.  C ) ) }
2217, 21eleqtrrdi 2234 . . 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 419 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  U. A  e.  U. ( B  X.  C
) )
241, 23sylbi 120 1  |-  ( A  e.  ( B  X.  C )  ->  U. A  e.  U. ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1332   E.wex 1469    e. wcel 1481   {cab 2126   {crab 2421   _Vcvv 2689   U.cuni 3743    X. cxp 4544   ` cfv 5130   1stc1st 6043   2ndc2nd 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fo 5136  df-fv 5138  df-1st 6045  df-2nd 6046
This theorem is referenced by: (None)
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