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Theorem xpimasn 4845
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 3541 . . 3  |-  ( X  e.  A  ->  E. x  x  e.  { X } )
2 snssi 3564 . . . . . 6  |-  ( X  e.  A  ->  { X }  C_  A )
3 dfss1 3193 . . . . . 6  |-  ( { X }  C_  A  <->  ( A  i^i  { X } )  =  { X } )
42, 3sylib 120 . . . . 5  |-  ( X  e.  A  ->  ( A  i^i  { X }
)  =  { X } )
54eleq2d 2154 . . . 4  |-  ( X  e.  A  ->  (
x  e.  ( A  i^i  { X }
)  <->  x  e.  { X } ) )
65exbidv 1750 . . 3  |-  ( X  e.  A  ->  ( E. x  x  e.  ( A  i^i  { X } )  <->  E. x  x  e.  { X } ) )
71, 6mpbird 165 . 2  |-  ( X  e.  A  ->  E. x  x  e.  ( A  i^i  { X } ) )
8 xpima2m 4844 . 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 1287   E.wex 1424    e. wcel 1436    i^i cin 2987    C_ wss 2988   {csn 3431    X. cxp 4409   "cima 4414
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3932  ax-pow 3984  ax-pr 4010
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ral 2360  df-rex 2361  df-v 2617  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-xp 4417  df-rel 4418  df-cnv 4419  df-dm 4421  df-rn 4422  df-res 4423  df-ima 4424
This theorem is referenced by: (None)
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