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Theorem fconstg 5220
Description: A cross product with a singleton is a constant function. (Contributed by NM, 19-Oct-2004.)
Assertion
Ref Expression
fconstg  |-  ( B  e.  V  ->  ( A  X.  { B }
) : A --> { B } )

Proof of Theorem fconstg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sneq 3461 . . . 4  |-  ( x  =  B  ->  { x }  =  { B } )
21xpeq2d 4475 . . 3  |-  ( x  =  B  ->  ( A  X.  { x }
)  =  ( A  X.  { B }
) )
3 feq1 5158 . . . 4  |-  ( ( A  X.  { x } )  =  ( A  X.  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { x } ) )
4 feq3 5160 . . . 4  |-  ( { x }  =  { B }  ->  ( ( A  X.  { B } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
53, 4sylan9bb 451 . . 3  |-  ( ( ( A  X.  {
x } )  =  ( A  X.  { B } )  /\  {
x }  =  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
62, 1, 5syl2anc 404 . 2  |-  ( x  =  B  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
7 vex 2623 . . 3  |-  x  e. 
_V
87fconst 5219 . 2  |-  ( A  X.  { x }
) : A --> { x }
96, 8vtoclg 2680 1  |-  ( B  e.  V  ->  ( A  X.  { B }
) : A --> { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1290    e. wcel 1439   {csn 3450    X. cxp 4449   -->wf 5024
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-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-fun 5030  df-fn 5031  df-f 5032
This theorem is referenced by:  fnconstg  5221  fconst6g  5222  xpsng  5486  fvconst2g  5525  fconst2g  5526
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