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Theorem xpimasn 5095
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 3725 . . 3  |-  ( X  e.  A  ->  E. x  x  e.  { X } )
2 snssi 3751 . . . . . 6  |-  ( X  e.  A  ->  { X }  C_  A )
3 dfss1 3354 . . . . . 6  |-  ( { X }  C_  A  <->  ( A  i^i  { X } )  =  { X } )
42, 3sylib 122 . . . . 5  |-  ( X  e.  A  ->  ( A  i^i  { X }
)  =  { X } )
54eleq2d 2259 . . . 4  |-  ( X  e.  A  ->  (
x  e.  ( A  i^i  { X }
)  <->  x  e.  { X } ) )
65exbidv 1836 . . 3  |-  ( X  e.  A  ->  ( E. x  x  e.  ( A  i^i  { X } )  <->  E. x  x  e.  { X } ) )
71, 6mpbird 167 . 2  |-  ( X  e.  A  ->  E. x  x  e.  ( A  i^i  { X } ) )
8 xpima2m 5094 . 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 1364   E.wex 1503    e. wcel 2160    i^i cin 3143    C_ wss 3144   {csn 3607    X. cxp 4642   "cima 4647
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-rel 4651  df-cnv 4652  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657
This theorem is referenced by:  imasnopn  14276
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