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Theorem fcoconst 6069
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 754 . . 3  |-  ( ( ( F  Fn  X  /\  Y  e.  X
)  /\  x  e.  I )  ->  Y  e.  X )
2 fconstmpt 5049 . . . 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 457 . . . . 5  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  Fn  X )
5 dffn2 5738 . . . . 5  |-  ( F  Fn  X  <->  F : X
--> _V )
64, 5sylib 196 . . . 4  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F : X --> _V )
76feqmptd 5927 . . 3  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  =  ( y  e.  X  |->  ( F `
 y ) ) )
8 fveq2 5872 . . 3  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
91, 3, 7, 8fmptco 6065 . 2  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  (
I  X.  { Y } ) )  =  ( x  e.  I  |->  ( F `  Y
) ) )
10 fconstmpt 5049 . 2  |-  ( I  X.  { ( F `
 Y ) } )  =  ( x  e.  I  |->  ( F `
 Y ) )
119, 10syl6eqr 2526 1  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  (
I  X.  { Y } ) )  =  ( I  X.  {
( F `  Y
) } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3118   {csn 4033    |-> cmpt 4511    X. cxp 5003    o. ccom 5009    Fn wfn 5589   -->wf 5590   ` cfv 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fv 5602
This theorem is referenced by:  s1co  12778  setcmon  15288  pwsco2mhm  15873  pws1  17135  pwsmgp  17137  imasdsf1olem  20742  cvmliftphtlem  28594  cvmlift3lem9  28604
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