Theorem fcoconst 5468

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 497	. . 3 |- ( ( ( F Fn X /\ Y e. X ) /\ x e. I ) -> Y e. X )
2		fconstmpt 4485	. . . 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 107	. . . . 5 |- ( ( F Fn X /\ Y e. X ) -> F Fn X )
5		dffn2 5163	. . . . 5 |- ( F Fn X <-> F : X --> _V )
6	4, 5	sylib 120	. . . 4 |- ( ( F Fn X /\ Y e. X ) -> F : X --> _V )
7	6	feqmptd 5357	. . 3 |- ( ( F Fn X /\ Y e. X ) -> F = ( y e. X |-> ( F ` y ) ) )
8		fveq2 5305	. . 3 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
9	1, 3, 7, 8	fmptco 5464	. 2 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( x e. I |-> ( F ` Y ) ) )
10		fconstmpt 4485	. 2 |- ( I X. { ( F ` Y ) } ) = ( x e. I |-> ( F ` Y ) )
11	9, 10	syl6eqr 2138	1 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( I X. { ( F ` Y ) } ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   _Vcvv 2619   {csn 3446   |-> cmpt 3899   X. cxp 4436   o. ccom 4442   Fn wfn 5010   -->wf 5011   ` cfv 5015

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fv 5023

This theorem is referenced by: (None)