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Theorem fcoconst 5708
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 528 . . 3  |-  ( ( ( F  Fn  X  /\  Y  e.  X
)  /\  x  e.  I )  ->  Y  e.  X )
2 fconstmpt 4691 . . . 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 109 . . . . 5  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  Fn  X )
5 dffn2 5386 . . . . 5  |-  ( F  Fn  X  <->  F : X
--> _V )
64, 5sylib 122 . . . 4  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F : X --> _V )
76feqmptd 5590 . . 3  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  F  =  ( y  e.  X  |->  ( F `
 y ) ) )
8 fveq2 5534 . . 3  |-  ( y  =  Y  ->  ( F `  y )  =  ( F `  Y ) )
91, 3, 7, 8fmptco 5703 . 2  |-  ( ( F  Fn  X  /\  Y  e.  X )  ->  ( F  o.  (
I  X.  { Y } ) )  =  ( x  e.  I  |->  ( F `  Y
) ) )
10 fconstmpt 4691 . 2  |-  ( I  X.  { ( F `
 Y ) } )  =  ( x  e.  I  |->  ( F `
 Y ) )
119, 10eqtr4di 2240 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 104    = wceq 1364    e. wcel 2160   _Vcvv 2752   {csn 3607    |-> cmpt 4079    X. cxp 4642    o. ccom 4648    Fn wfn 5230   -->wf 5231   ` cfv 5235
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243
This theorem is referenced by: (None)
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