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Theorem fcoconst 5908
Description: Composition with a constant function. (Contributed by Stefan O'Rear, 11-Mar-2015.)
Assertion
Ref Expression
fcoconst  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  (
I  X.  { Y } ) )  =  ( I  X.  {
( F `  Y
) } ) )

Proof of Theorem fcoconst
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 733 . . 3  |-  ( ( ( F  Fn  X  /\  Y  e.  X
)  /\  x  e.  I )  ->  Y  e.  X )
2 fconstmpt 4924 . . . 4  |-  ( I  X.  { Y }
)  =  ( x  e.  I  |->  Y )
32a1i 11 . . 3  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( I  X.  { Y } )  =  ( x  e.  I  |->  Y ) )
4 simpl 445 . . . . 5  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  Fn  X )
5 dffn2 5595 . . . . 5  |-  ( F  Fn  X  <->  F : X
--> _V )
64, 5sylib 190 . . . 4  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F : X --> _V )
76feqmptd 5782 . . 3  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  =  ( y  e.  X  |->  ( F `
 y ) ) )
8 fveq2 5731 . . 3  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
91, 3, 7, 8fmptco 5904 . 2  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  (
I  X.  { Y } ) )  =  ( x  e.  I  |->  ( F `  Y
) ) )
10 fconstmpt 4924 . 2  |-  ( I  X.  { ( F `
 Y ) } )  =  ( x  e.  I  |->  ( F `
 Y ) )
119, 10syl6eqr 2488 1  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  (
I  X.  { Y } ) )  =  ( I  X.  {
( F `  Y
) } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816    e. cmpt 4269    X. cxp 4879    o. ccom 4885    Fn wfn 5452   -->wf 5453   ` cfv 5457
This theorem is referenced by:  s1co  11807  setcmon  14247  pwsco2mhm  14775  pws1  15727  pwsmgp  15729  imasdsf1olem  18408  cvmliftphtlem  25009  cvmlift3lem9  25019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465
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