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Theorem xpimasn 4797
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 3514 . . 3  |-  ( X  e.  A  ->  E. x  x  e.  { X } )
2 snssi 3536 . . . . . 6  |-  ( X  e.  A  ->  { X }  C_  A )
3 dfss1 3169 . . . . . 6  |-  ( { X }  C_  A  <->  ( A  i^i  { X } )  =  { X } )
42, 3sylib 131 . . . . 5  |-  ( X  e.  A  ->  ( A  i^i  { X }
)  =  { X } )
54eleq2d 2123 . . . 4  |-  ( X  e.  A  ->  (
x  e.  ( A  i^i  { X }
)  <->  x  e.  { X } ) )
65exbidv 1722 . . 3  |-  ( X  e.  A  ->  ( E. x  x  e.  ( A  i^i  { X } )  <->  E. x  x  e.  { X } ) )
71, 6mpbird 160 . 2  |-  ( X  e.  A  ->  E. x  x  e.  ( A  i^i  { X } ) )
8 xpima2m 4796 . 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 1259   E.wex 1397    e. wcel 1409    i^i cin 2944    C_ wss 2945   {csn 3403    X. cxp 4371   "cima 4376
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386
This theorem is referenced by: (None)
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