MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fconstg Unicode version

Theorem fconstg 5444
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 3664 . . . 4  |-  ( x  =  B  ->  { x }  =  { B } )
21xpeq2d 4729 . . 3  |-  ( x  =  B  ->  ( A  X.  { x }
)  =  ( A  X.  { B }
) )
3 feq1 5391 . . . 4  |-  ( ( A  X.  { x } )  =  ( A  X.  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { x } ) )
4 feq3 5393 . . . 4  |-  ( { x }  =  { B }  ->  ( ( A  X.  { B } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
53, 4sylan9bb 680 . . 3  |-  ( ( ( A  X.  {
x } )  =  ( A  X.  { B } )  /\  {
x }  =  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
62, 1, 5syl2anc 642 . 2  |-  ( x  =  B  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
7 vex 2804 . . 3  |-  x  e. 
_V
87fconst 5443 . 2  |-  ( A  X.  { x }
) : A --> { x }
96, 8vtoclg 2856 1  |-  ( B  e.  V  ->  ( A  X.  { B }
) : A --> { B } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {csn 3653    X. cxp 4703   -->wf 5267
This theorem is referenced by:  fnconstg  5445  fconst6g  5446  xpsng  5715  fvconst2g  5743  fconst2g  5744  xkoptsub  17364  mbfconstlem  19000  i1fmulclem  19073  i1fmulc  19074  itg2mulclem  19117  dvcmulf  19310  dvef  19343  coemulc  19652  itg2addnclem  25003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-fun 5273  df-fn 5274  df-f 5275
  Copyright terms: Public domain W3C validator