Theorem fcoconst 5690

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 528	. . 3 |- ( ( ( F Fn X /\ Y e. X ) /\ x e. I ) -> Y e. X )
2		fconstmpt 4675	. . . 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 5369	. . . . 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 5572	. . 3 |- ( ( F Fn X /\ Y e. X ) -> F = ( y e. X |-> ( F ` y ) ) )
8		fveq2 5517	. . 3 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
9	1, 3, 7, 8	fmptco 5685	. 2 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( x e. I |-> ( F ` Y ) ) )
10		fconstmpt 4675	. 2 |- ( I X. { ( F ` Y ) } ) = ( x e. I |-> ( F ` Y ) )
11	9, 10	eqtr4di 2228	1 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( I X. { ( F ` Y ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   _Vcvv 2739   {csn 3594   |-> cmpt 4066   X. cxp 4626   o. ccom 4632   Fn wfn 5213   -->wf 5214   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fv 5226

This theorem is referenced by: (None)