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Theorem fconst 5288
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 4556 . . 3  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
31, 2fnmpti 5221 . 2  |-  ( A  X.  { B }
)  Fn  A
4 rnxpss 4940 . 2  |-  ran  ( A  X.  { B }
)  C_  { B }
5 df-f 5097 . 2  |-  ( ( A  X.  { B } ) : A --> { B }  <->  ( ( A  X.  { B }
)  Fn  A  /\  ran  ( A  X.  { B } )  C_  { B } ) )
63, 4, 5mpbir2an 911 1  |-  ( A  X.  { B }
) : A --> { B }
Colors of variables: wff set class
Syntax hints:    e. wcel 1465   _Vcvv 2660    C_ wss 3041   {csn 3497    X. cxp 4507   ran crn 4510    Fn wfn 5088   -->wf 5089
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097
This theorem is referenced by:  fconstg  5289  exmidfodomrlemim  7025  ser0f  10256  dvexp  12771
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