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Theorem xpimasn 5047
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 3689 . . 3  |-  ( X  e.  A  ->  E. x  x  e.  { X } )
2 snssi 3712 . . . . . 6  |-  ( X  e.  A  ->  { X }  C_  A )
3 dfss1 3322 . . . . . 6  |-  ( { X }  C_  A  <->  ( A  i^i  { X } )  =  { X } )
42, 3sylib 121 . . . . 5  |-  ( X  e.  A  ->  ( A  i^i  { X }
)  =  { X } )
54eleq2d 2234 . . . 4  |-  ( X  e.  A  ->  (
x  e.  ( A  i^i  { X }
)  <->  x  e.  { X } ) )
65exbidv 1812 . . 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 5046 . 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 1342   E.wex 1479    e. wcel 2135    i^i cin 3111    C_ wss 3112   {csn 3571    X. cxp 4597   "cima 4602
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4095  ax-pow 4148  ax-pr 4182
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2724  df-un 3116  df-in 3118  df-ss 3125  df-pw 3556  df-sn 3577  df-pr 3578  df-op 3580  df-br 3978  df-opab 4039  df-xp 4605  df-rel 4606  df-cnv 4607  df-dm 4609  df-rn 4610  df-res 4611  df-ima 4612
This theorem is referenced by:  imasnopn  12857
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