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Theorem fcoconst 5897
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 732 . . 3  |-  ( ( ( F  Fn  X  /\  Y  e.  X
)  /\  x  e.  I )  ->  Y  e.  X )
2 fconstmpt 4913 . . . 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 444 . . . . 5  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  Fn  X )
5 dffn2 5584 . . . . 5  |-  ( F  Fn  X  <->  F : X
--> _V )
64, 5sylib 189 . . . 4  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F : X --> _V )
76feqmptd 5771 . . 3  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  =  ( y  e.  X  |->  ( F `
 y ) ) )
8 fveq2 5720 . . 3  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
91, 3, 7, 8fmptco 5893 . 2  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  (
I  X.  { Y } ) )  =  ( x  e.  I  |->  ( F `  Y
) ) )
10 fconstmpt 4913 . 2  |-  ( I  X.  { ( F `
 Y ) } )  =  ( x  e.  I  |->  ( F `
 Y ) )
119, 10syl6eqr 2485 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 359    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806    e. cmpt 4258    X. cxp 4868    o. ccom 4874    Fn wfn 5441   -->wf 5442   ` cfv 5446
This theorem is referenced by:  s1co  11794  setcmon  14234  pwsco2mhm  14762  pws1  15714  pwsmgp  15716  imasdsf1olem  18395  cvmliftphtlem  24996  cvmlift3lem9  25006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454
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