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Theorem fconst 5107
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 4407 . . 3  |-  ( A  X.  { B }
)  =  ( x  e.  A  |->  B )
31, 2fnmpti 5052 . 2  |-  ( A  X.  { B }
)  Fn  A
4 rnxpss 4778 . 2  |-  ran  ( A  X.  { B }
)  C_  { B }
5 df-f 4930 . 2  |-  ( ( A  X.  { B } ) : A --> { B }  <->  ( ( A  X.  { B }
)  Fn  A  /\  ran  ( A  X.  { B } )  C_  { B } ) )
63, 4, 5mpbir2an 884 1  |-  ( A  X.  { B }
) : A --> { B }
Colors of variables: wff set class
Syntax hints:    e. wcel 1434   _Vcvv 2602    C_ wss 2974   {csn 3400    X. cxp 4363   ran crn 4366    Fn wfn 4921   -->wf 4922
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-br 3788  df-opab 3842  df-mpt 3843  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-fun 4928  df-fn 4929  df-f 4930
This theorem is referenced by:  fconstg  5108  iser0f  9558
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