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Theorem fconstg 5778
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 4042 . . . 4  |-  ( x  =  B  ->  { x }  =  { B } )
21xpeq2d 5032 . . 3  |-  ( x  =  B  ->  ( A  X.  { x }
)  =  ( A  X.  { B }
) )
3 feq1 5719 . . . 4  |-  ( ( A  X.  { x } )  =  ( A  X.  { B } )  ->  (
( A  X.  {
x } ) : A --> { x }  <->  ( A  X.  { B } ) : A --> { x } ) )
4 feq3 5721 . . . 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 3112 . . 3  |-  x  e. 
_V
87fconst 5777 . 2  |-  ( A  X.  { x }
) : A --> { x }
96, 8vtoclg 3167 1  |-  ( B  e.  V  ->  ( A  X.  { B }
) : A --> { B } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   {csn 4032    X. cxp 5006   -->wf 5590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-fun 5596  df-fn 5597  df-f 5598
This theorem is referenced by:  fnconstg  5779  fconst6g  5780  xpsng  6073  fvconst2g  6126  fconst2g  6127  xkoptsub  20280  mbfconstlem  22161  i1fmulclem  22234  i1fmulc  22235  itg2mulclem  22278  dvcmulf  22473  dvef  22506  coemulc  22777  resf1o  27703  locfinref  27997  ccatmulgnn0dir  28671
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