Theorem fcoconst 5355

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 496	. . 3 |- ( ( ( F Fn X /\ Y e. X ) /\ x e. I ) -> Y e. X )
2		fconstmpt 4405	. . . 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 5067	. . . . 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 5247	. . 3 |- ( ( F Fn X /\ Y e. X ) -> F = ( y e. X |-> ( F ` y ) ) )
8		fveq2 5198	. . 3 |- ( y = Y -> ( F ` y ) = ( F ` Y ) )
9	1, 3, 7, 8	fmptco 5351	. 2 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( x e. I |-> ( F ` Y ) ) )
10		fconstmpt 4405	. 2 |- ( I X. { ( F ` Y ) } ) = ( x e. I |-> ( F ` Y ) )
11	9, 10	syl6eqr 2131	1 |- ( ( F Fn X /\ Y e. X ) -> ( F o. ( I X. { Y } ) ) = ( I X. { ( F ` Y ) } ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   _Vcvv 2601   {csn 3398   |-> cmpt 3839   X. cxp 4361   o. ccom 4367   Fn wfn 4917   -->wf 4918   ` cfv 4922

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-fv 4930

This theorem is referenced by: (None)