Theorem fcoconst 5591

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.

Step	Hyp	Ref	Expression
1		simplr 519	. . 3 |- ( ( ( F Fn X /\ Y e. X ) /\ x e. I ) -> Y e. X )
2		fconstmpt 4586	. . . 4 |- ( I X. { Y } ) = ( x e. I |-> Y )
3	2	a1i 9	. . 3 |- ( ( F Fn X /\ Y e. X ) -> ( I X. { Y } ) = ( x e. I |-> Y ) )
4		simpl 108	. . . . 5 |- ( ( F Fn X /\ Y e. X ) -> F Fn X )
5		dffn2 5274	. . . . 5 |- ( F Fn X <-> F : X --> _V )
6	4, 5	sylib 121	. . . 4 |- ( ( F Fn X /\ Y e. X ) -> F : X --> _V )
7	6	feqmptd 5474	. . 3 |- ( ( F Fn X /\ Y e. X ) -> F = ( y e. X |-> ( F ` y ) ) )
8		fveq2 5421	. . 3 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
9	1, 3, 7, 8	fmptco 5586	. 2 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( x e. I |-> ( F ` Y ) ) )
10		fconstmpt 4586	. 2 |- ( I X. { ( F ` Y ) } ) = ( x e. I |-> ( F ` Y ) )
11	9, 10	syl6eqr 2190	1 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( I X. { ( F ` Y ) } ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   _Vcvv 2686   {csn 3527   |-> cmpt 3989   X. cxp 4537   o. ccom 4543   Fn wfn 5118   -->wf 5119   ` cfv 5123

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131

This theorem is referenced by: (None)