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Theorem rnxpid 4742
Description: The range of a square cross product. (Contributed by FL, 17-May-2010.)
Assertion
Ref Expression
rnxpid  |-  ran  ( A  X.  A )  =  A

Proof of Theorem rnxpid
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rnxpss 4741 . 2  |-  ran  ( A  X.  A )  C_  A
2 opelxp 4361 . . . . . 6  |-  ( <.
x ,  x >.  e.  ( A  X.  A
)  <->  ( x  e.  A  /\  x  e.  A ) )
3 anidm 376 . . . . . 6  |-  ( ( x  e.  A  /\  x  e.  A )  <->  x  e.  A )
42, 3bitri 173 . . . . 5  |-  ( <.
x ,  x >.  e.  ( A  X.  A
)  <->  x  e.  A
)
5 opeq1 3546 . . . . . . . . 9  |-  ( x  =  y  ->  <. x ,  x >.  =  <. y ,  x >. )
65eleq1d 2106 . . . . . . . 8  |-  ( x  =  y  ->  ( <. x ,  x >.  e.  ( A  X.  A
)  <->  <. y ,  x >.  e.  ( A  X.  A ) ) )
76equcoms 1594 . . . . . . 7  |-  ( y  =  x  ->  ( <. x ,  x >.  e.  ( A  X.  A
)  <->  <. y ,  x >.  e.  ( A  X.  A ) ) )
87biimpd 132 . . . . . 6  |-  ( y  =  x  ->  ( <. x ,  x >.  e.  ( A  X.  A
)  ->  <. y ,  x >.  e.  ( A  X.  A ) ) )
98spimev 1741 . . . . 5  |-  ( <.
x ,  x >.  e.  ( A  X.  A
)  ->  E. y <. y ,  x >.  e.  ( A  X.  A
) )
104, 9sylbir 125 . . . 4  |-  ( x  e.  A  ->  E. y <. y ,  x >.  e.  ( A  X.  A
) )
11 vex 2557 . . . . 5  |-  x  e. 
_V
1211elrn2 4563 . . . 4  |-  ( x  e.  ran  ( A  X.  A )  <->  E. y <. y ,  x >.  e.  ( A  X.  A
) )
1310, 12sylibr 137 . . 3  |-  ( x  e.  A  ->  x  e.  ran  ( A  X.  A ) )
1413ssriv 2946 . 2  |-  A  C_  ran  ( A  X.  A
)
151, 14eqssi 2958 1  |-  ran  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 97    <-> wb 98    = wceq 1243   E.wex 1381    e. wcel 1393   <.cop 3375    X. cxp 4330   ran crn 4333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3872  ax-pow 3924  ax-pr 3941
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2308  df-rex 2309  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-br 3762  df-opab 3816  df-xp 4338  df-rel 4339  df-cnv 4340  df-dm 4342  df-rn 4343
This theorem is referenced by: (None)
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