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Theorem elxp7 6146
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5096. (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 elex 2741 . 2  |-  ( A  e.  ( B  X.  C )  ->  A  e.  _V )
2 elex 2741 . . 3  |-  ( A  e.  ( _V  X.  _V )  ->  A  e. 
_V )
32adantr 274 . 2  |-  ( ( A  e.  ( _V 
X.  _V )  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) )  ->  A  e.  _V )
4 elxp6 6145 . . 3  |-  ( A  e.  ( B  X.  C )  <->  ( A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  B  /\  ( 2nd `  A )  e.  C ) ) )
5 elxp6 6145 . . . . 5  |-  ( A  e.  ( _V  X.  _V )  <->  ( A  = 
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  /\  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) ) )
6 1stexg 6143 . . . . . . 7  |-  ( A  e.  _V  ->  ( 1st `  A )  e. 
_V )
7 2ndexg 6144 . . . . . . 7  |-  ( A  e.  _V  ->  ( 2nd `  A )  e. 
_V )
86, 7jca 304 . . . . . 6  |-  ( A  e.  _V  ->  (
( 1st `  A
)  e.  _V  /\  ( 2nd `  A )  e.  _V ) )
98biantrud 302 . . . . 5  |-  ( A  e.  _V  ->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A )
>. 
<->  ( A  =  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  /\  ( ( 1st `  A )  e. 
_V  /\  ( 2nd `  A )  e.  _V ) ) ) )
105, 9bitr4id 198 . . . 4  |-  ( A  e.  _V  ->  ( A  e.  ( _V  X.  _V )  <->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
)
1110anbi1d 462 . . 3  |-  ( A  e.  _V  ->  (
( 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
) ) ) )
124, 11bitr4id 198 . 2  |-  ( A  e.  _V  ->  ( A  e.  ( B  X.  C )  <->  ( A  e.  ( _V  X.  _V )  /\  ( ( 1st `  A )  e.  B  /\  ( 2nd `  A
)  e.  C ) ) ) )
131, 3, 12pm5.21nii 699 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:    /\ wa 103    <-> wb 104    = wceq 1348    e. wcel 2141   _Vcvv 2730   <.cop 3584    X. cxp 4607   ` cfv 5196   1stc1st 6114   2ndc2nd 6115
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-fo 5202  df-fv 5204  df-1st 6116  df-2nd 6117
This theorem is referenced by:  xp2  6149  unielxp  6150  1stconst  6197  2ndconst  6198  f1od2  6211
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