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Theorem fconstg 5618
Description: A Cartesian 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 3908 . . . 4  |-  ( x  =  B  ->  { x }  =  { B } )
21xpeq2d 4885 . . 3  |-  ( x  =  B  ->  ( A  X.  { x }
)  =  ( A  X.  { B }
) )
3 feq1 5563 . . . 4  |-  ( ( A  X.  { x } )  =  ( A  X.  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { x } ) )
4 feq3 5565 . . . 4  |-  ( { x }  =  { B }  ->  ( ( A  X.  { B } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
53, 4sylan9bb 699 . . 3  |-  ( ( ( A  X.  {
x } )  =  ( A  X.  { B } )  /\  {
x }  =  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
62, 1, 5syl2anc 661 . 2  |-  ( x  =  B  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { B } ) )
7 vex 2996 . . 3  |-  x  e. 
_V
87fconst 5617 . 2  |-  ( A  X.  { x }
) : A --> { x }
96, 8vtoclg 3051 1  |-  ( B  e.  V  ->  ( A  X.  { B }
) : A --> { B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   {csn 3898    X. cxp 4859   -->wf 5435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pr 4552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2741  df-rex 2742  df-rab 2745  df-v 2995  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-sn 3899  df-pr 3901  df-op 3905  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-fun 5441  df-fn 5442  df-f 5443
This theorem is referenced by:  fnconstg  5619  fconst6g  5620  xpsng  5905  fvconst2g  5952  fconst2g  5953  xkoptsub  19249  mbfconstlem  21129  i1fmulclem  21202  i1fmulc  21203  itg2mulclem  21246  dvcmulf  21441  dvef  21474  coemulc  21744  resf1o  26052  ccatmulgnn0dir  26962
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