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Theorem xpimasn 4892
Description: The image of a singleton by a cross product. (Contributed by Thierry Arnoux, 14-Jan-2018.)
Assertion
Ref Expression
xpimasn  |-  ( X  e.  A  ->  (
( A  X.  B
) " { X } )  =  B )

Proof of Theorem xpimasn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 snmg 3564 . . 3  |-  ( X  e.  A  ->  E. x  x  e.  { X } )
2 snssi 3587 . . . . . 6  |-  ( X  e.  A  ->  { X }  C_  A )
3 dfss1 3205 . . . . . 6  |-  ( { X }  C_  A  <->  ( A  i^i  { X } )  =  { X } )
42, 3sylib 121 . . . . 5  |-  ( X  e.  A  ->  ( A  i^i  { X }
)  =  { X } )
54eleq2d 2158 . . . 4  |-  ( X  e.  A  ->  (
x  e.  ( A  i^i  { X }
)  <->  x  e.  { X } ) )
65exbidv 1754 . . 3  |-  ( X  e.  A  ->  ( E. x  x  e.  ( A  i^i  { X } )  <->  E. x  x  e.  { X } ) )
71, 6mpbird 166 . 2  |-  ( X  e.  A  ->  E. x  x  e.  ( A  i^i  { X } ) )
8 xpima2m 4891 . 2  |-  ( E. x  x  e.  ( A  i^i  { X } )  ->  (
( A  X.  B
) " { X } )  =  B )
97, 8syl 14 1  |-  ( X  e.  A  ->  (
( A  X.  B
) " { X } )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1290   E.wex 1427    e. wcel 1439    i^i cin 2999    C_ wss 3000   {csn 3450    X. cxp 4449   "cima 4454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-br 3852  df-opab 3906  df-xp 4457  df-rel 4458  df-cnv 4459  df-dm 4461  df-rn 4462  df-res 4463  df-ima 4464
This theorem is referenced by: (None)
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