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Theorem rnxpid 4865
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 4862 . 2  |-  ran  ( A  X.  A )  C_  A
2 opelxp 4467 . . . . . 6  |-  ( <.
x ,  x >.  e.  ( A  X.  A
)  <->  ( x  e.  A  /\  x  e.  A ) )
3 anidm 388 . . . . . 6  |-  ( ( x  e.  A  /\  x  e.  A )  <->  x  e.  A )
42, 3bitri 182 . . . . 5  |-  ( <.
x ,  x >.  e.  ( A  X.  A
)  <->  x  e.  A
)
5 opeq1 3622 . . . . . . . . 9  |-  ( x  =  y  ->  <. x ,  x >.  =  <. y ,  x >. )
65eleq1d 2156 . . . . . . . 8  |-  ( x  =  y  ->  ( <. x ,  x >.  e.  ( A  X.  A
)  <->  <. y ,  x >.  e.  ( A  X.  A ) ) )
76equcoms 1641 . . . . . . 7  |-  ( y  =  x  ->  ( <. x ,  x >.  e.  ( A  X.  A
)  <->  <. y ,  x >.  e.  ( A  X.  A ) ) )
87biimpd 142 . . . . . 6  |-  ( y  =  x  ->  ( <. x ,  x >.  e.  ( A  X.  A
)  ->  <. y ,  x >.  e.  ( A  X.  A ) ) )
98spimev 1789 . . . . 5  |-  ( <.
x ,  x >.  e.  ( A  X.  A
)  ->  E. y <. y ,  x >.  e.  ( A  X.  A
) )
104, 9sylbir 133 . . . 4  |-  ( x  e.  A  ->  E. y <. y ,  x >.  e.  ( A  X.  A
) )
11 vex 2622 . . . . 5  |-  x  e. 
_V
1211elrn2 4677 . . . 4  |-  ( x  e.  ran  ( A  X.  A )  <->  E. y <. y ,  x >.  e.  ( A  X.  A
) )
1310, 12sylibr 132 . . 3  |-  ( x  e.  A  ->  x  e.  ran  ( A  X.  A ) )
1413ssriv 3029 . 2  |-  A  C_  ran  ( A  X.  A
)
151, 14eqssi 3041 1  |-  ran  ( A  X.  A )  =  A
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   <.cop 3449    X. cxp 4436   ran crn 4439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-rel 4445  df-cnv 4446  df-dm 4448  df-rn 4449
This theorem is referenced by: (None)
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