ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fconstg Unicode version

Theorem fconstg 5319
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 3538 . . . 4  |-  ( x  =  B  ->  { x }  =  { B } )
21xpeq2d 4563 . . 3  |-  ( x  =  B  ->  ( A  X.  { x }
)  =  ( A  X.  { B }
) )
3 feq1 5255 . . . 4  |-  ( ( A  X.  { x } )  =  ( A  X.  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { x } ) )
4 feq3 5257 . . . 4  |-  ( { x }  =  { B }  ->  ( ( A  X.  { B } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
53, 4sylan9bb 457 . . 3  |-  ( ( ( A  X.  {
x } )  =  ( A  X.  { B } )  /\  {
x }  =  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
62, 1, 5syl2anc 408 . 2  |-  ( x  =  B  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
7 vex 2689 . . 3  |-  x  e. 
_V
87fconst 5318 . 2  |-  ( A  X.  { x }
) : A --> { x }
96, 8vtoclg 2746 1  |-  ( B  e.  V  ->  ( A  X.  { B }
) : A --> { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   {csn 3527    X. cxp 4537   -->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:  fnconstg  5320  fconst6g  5321  xpsng  5595  fvconst2g  5634  fconst2g  5635  dvef  12871
  Copyright terms: Public domain W3C validator