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Theorem fcoconst 5386
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 497 . . 3  |-  ( ( ( F  Fn  X  /\  Y  e.  X
)  /\  x  e.  I )  ->  Y  e.  X )
2 fconstmpt 4433 . . . 4  |-  ( I  X.  { Y }
)  =  ( x  e.  I  |->  Y )
32a1i 9 . . 3  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( I  X.  { Y } )  =  ( x  e.  I  |->  Y ) )
4 simpl 107 . . . . 5  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  Fn  X )
5 dffn2 5098 . . . . 5  |-  ( F  Fn  X  <->  F : X
--> _V )
64, 5sylib 120 . . . 4  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F : X --> _V )
76feqmptd 5278 . . 3  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  =  ( y  e.  X  |->  ( F `
 y ) ) )
8 fveq2 5229 . . 3  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
91, 3, 7, 8fmptco 5382 . 2  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  (
I  X.  { Y } ) )  =  ( x  e.  I  |->  ( F `  Y
) ) )
10 fconstmpt 4433 . 2  |-  ( I  X.  { ( F `
 Y ) } )  =  ( x  e.  I  |->  ( F `
 Y ) )
119, 10syl6eqr 2133 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 102    = wceq 1285    e. wcel 1434   _Vcvv 2610   {csn 3416    |-> cmpt 3859    X. cxp 4389    o. ccom 4395    Fn wfn 4947   -->wf 4948   ` cfv 4952
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-fv 4960
This theorem is referenced by: (None)
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