Theorem offres 6041

Description: Pointwise combination commutes with restriction. (Contributed by Stefan O'Rear, 24-Jan-2015.)

Assertion

Ref	Expression
offres	|- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) = ( ( F |` D ) oF R ( G |` D ) ) )

Proof of Theorem offres

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss2 3302	. . . . . 6 |- ( ( dom F i^i dom G ) i^i D ) C_ D
2	1	sseli 3098	. . . . 5 |- ( x e. ( ( dom F i^i dom G ) i^i D ) -> x e. D )
3		fvres 5453	. . . . . 6 |- ( x e. D -> ( ( F |` D ) ` x ) = ( F ` x ) )
4		fvres 5453	. . . . . 6 |- ( x e. D -> ( ( G |` D ) ` x ) = ( G ` x ) )
5	3, 4	oveq12d 5800	. . . . 5 |- ( x e. D -> ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
6	2, 5	syl 14	. . . 4 |- ( x e. ( ( dom F i^i dom G ) i^i D ) -> ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
7	6	mpteq2ia 4022	. . 3 |- ( x e. ( ( dom F i^i dom G ) i^i D ) |-> ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) ) = ( x e. ( ( dom F i^i dom G ) i^i D ) |-> ( ( F ` x ) R ( G ` x ) ) )
8		inindi 3298	. . . . 5 |- ( D i^i ( dom F i^i dom G ) ) = ( ( D i^i dom F ) i^i ( D i^i dom G ) )
9		incom 3273	. . . . 5 |- ( ( dom F i^i dom G ) i^i D ) = ( D i^i ( dom F i^i dom G ) )
10		dmres 4848	. . . . . 6 |- dom ( F |` D ) = ( D i^i dom F )
11		dmres 4848	. . . . . 6 |- dom ( G |` D ) = ( D i^i dom G )
12	10, 11	ineq12i 3280	. . . . 5 |- ( dom ( F |` D ) i^i dom ( G |` D ) ) = ( ( D i^i dom F ) i^i ( D i^i dom G ) )
13	8, 9, 12	3eqtr4ri 2172	. . . 4 |- ( dom ( F |` D ) i^i dom ( G |` D ) ) = ( ( dom F i^i dom G ) i^i D )
14		eqid 2140	. . . 4 |- ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) = ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) )
15	13, 14	mpteq12i 4024	. . 3 |- ( x e. ( dom ( F |` D ) i^i dom ( G |` D ) ) |-> ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) ) = ( x e. ( ( dom F i^i dom G ) i^i D ) |-> ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) )
16		resmpt3 4876	. . 3 |- ( ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) |` D ) = ( x e. ( ( dom F i^i dom G ) i^i D ) |-> ( ( F ` x ) R ( G ` x ) ) )
17	7, 15, 16	3eqtr4ri 2172	. 2 |- ( ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) |` D ) = ( x e. ( dom ( F |` D ) i^i dom ( G |` D ) ) |-> ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) )
18		offval3 6040	. . 3 |- ( ( F e. V /\ G e. W ) -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
19	18	reseq1d 4826	. 2 |- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) = ( ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) |` D ) )
20		resexg 4867	. . 3 |- ( F e. V -> ( F |` D ) e. _V )
21		resexg 4867	. . 3 |- ( G e. W -> ( G |` D ) e. _V )
22		offval3 6040	. . 3 |- ( ( ( F |` D ) e. _V /\ ( G |` D ) e. _V ) -> ( ( F |` D ) oF R ( G |` D ) ) = ( x e. ( dom ( F |` D ) i^i dom ( G |` D ) ) |-> ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) ) )
23	20, 21, 22	syl2an 287	. 2 |- ( ( F e. V /\ G e. W ) -> ( ( F |` D ) oF R ( G |` D ) ) = ( x e. ( dom ( F |` D ) i^i dom ( G |` D ) ) |-> ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) ) )
24	17, 19, 23	3eqtr4a 2199	1 |- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) = ( ( F |` D ) oF R ( G |` D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   i^i cin 3075   |-> cmpt 3997   dom cdm 4547   |` cres 4549   ` cfv 5131  (class class class)co 5782   oFcof 5988

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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-of 5990

This theorem is referenced by: (None)