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Theorem elxp7 6371
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5349. (Contributed by NM, 19-Aug-2006.)
Assertion
Ref Expression
elxp7  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )

Proof of Theorem elxp7
StepHypRef Expression
1 elxp6 6370 . 2  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
2 fvex 5734 . . . . 5  |-  ( 1st `  A )  e.  _V
3 fvex 5734 . . . . 5  |-  ( 2nd `  A )  e.  _V
42, 3pm3.2i 442 . . . 4  |-  ( ( 1st `  A )  e.  _V  /\  ( 2nd `  A )  e. 
_V )
5 elxp6 6370 . . . 4  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
64, 5mpbiran2 886 . . 3  |-  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >. )
76anbi1i 677 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  <-> 
( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A )  e.  C
) ) )
81, 7bitr4i 244 1  |-  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809    X. cxp 4868   ` cfv 5446   1stc1st 6339   2ndc2nd 6340
This theorem is referenced by:  xp2  6376  unielxp  6377  1stconst  6427  2ndconst  6428  fparlem1  6438  fparlem2  6439  infxpenlem  7887  1stpreima  24087  2ndpreima  24088  xpinpreima2  24297  tpr2rico  24302  sxbrsigalem0  24613  dya2iocnrect  24623  pellex  26879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fv 5454  df-1st 6341  df-2nd 6342
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