Theorem fcoconst 5818

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 529	. . 3 |- ( ( ( F Fn X /\ Y e. X ) /\ x e. I ) -> Y e. X )
2		fconstmpt 4773	. . . 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 109	. . . . 5 |- ( ( F Fn X /\ Y e. X ) -> F Fn X )
5		dffn2 5484	. . . . 5 |- ( F Fn X <-> F : X --> _V )
6	4, 5	sylib 122	. . . 4 |- ( ( F Fn X /\ Y e. X ) -> F : X --> _V )
7	6	feqmptd 5699	. . 3 |- ( ( F Fn X /\ Y e. X ) -> F = ( y e. X |-> ( F ` y ) ) )
8		fveq2 5639	. . 3 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
9	1, 3, 7, 8	fmptco 5813	. 2 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( x e. I |-> ( F ` Y ) ) )
10		fconstmpt 4773	. 2 |- ( I X. { ( F ` Y ) } ) = ( x e. I |-> ( F ` Y ) )
11	9, 10	eqtr4di 2282	1 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( I X. { ( F ` Y ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   {csn 3669   |-> cmpt 4150   X. cxp 4723   o. ccom 4729   Fn wfn 5321   -->wf 5322   ` cfv 5326

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334

This theorem is referenced by: (None)