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Theorem fconst 5318
Description: A cross product with a singleton is a constant function. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 17-Sep-2011.)
Hypothesis
Ref Expression
fconst.1  |-  B  e. 
_V
Assertion
Ref Expression
fconst  |-  ( A  X.  { B }
) : A --> { B }

Proof of Theorem fconst
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fconst.1 . . 3  |-  B  e. 
_V
2 fconstmpt 4586 . . 3  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
31, 2fnmpti 5251 . 2  |-  ( A  X.  { B }
)  Fn  A
4 rnxpss 4970 . 2  |-  ran  ( A  X.  { B }
)  C_  { B }
5 df-f 5127 . 2  |-  ( ( A  X.  { B } ) : A --> { B }  <->  ( ( A  X.  { B }
)  Fn  A  /\  ran  ( A  X.  { B } )  C_  { B } ) )
63, 4, 5mpbir2an 926 1  |-  ( A  X.  { B }
) : A --> { B }
Colors of variables: wff set class
Syntax hints:    e. wcel 1480   _Vcvv 2686    C_ wss 3071   {csn 3527    X. cxp 4537   ran crn 4540    Fn wfn 5118   -->wf 5119
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-fun 5125  df-fn 5126  df-f 5127
This theorem is referenced by:  fconstg  5319  exmidfodomrlemim  7057  ser0f  10295  prodf1f  11319  dvexp  12854
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