Theorem offres 6306

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 3430	. . . . . 6 |- ( ( dom F i^i dom G ) i^i D ) C_ D
2	1	sseli 3224	. . . . 5 |- ( x e. ( ( dom F i^i dom G ) i^i D ) -> x e. D )
3		fvres 5672	. . . . . 6 |- ( x e. D -> ( ( F |` D ) ` x ) = ( F ` x ) )
4		fvres 5672	. . . . . 6 |- ( x e. D -> ( ( G |` D ) ` x ) = ( G ` x ) )
5	3, 4	oveq12d 6046	. . . . 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 4180	. . 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 3426	. . . . 5 |- ( D i^i ( dom F i^i dom G ) ) = ( ( D i^i dom F ) i^i ( D i^i dom G ) )
9		incom 3401	. . . . 5 |- ( ( dom F i^i dom G ) i^i D ) = ( D i^i ( dom F i^i dom G ) )
10		dmres 5040	. . . . . 6 |- dom ( F |` D ) = ( D i^i dom F )
11		dmres 5040	. . . . . 6 |- dom ( G |` D ) = ( D i^i dom G )
12	10, 11	ineq12i 3408	. . . . 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 2263	. . . 4 |- ( dom ( F |` D ) i^i dom ( G |` D ) ) = ( ( dom F i^i dom G ) i^i D )
14		eqid 2231	. . . 4 |- ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) = ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) )
15	13, 14	mpteq12i 4182	. . 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 5068	. . 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 2263	. 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 6305	. . 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 5018	. 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 5059	. . 3 |- ( F e. V -> ( F |` D ) e. _V )
21		resexg 5059	. . 3 |- ( G e. W -> ( G |` D ) e. _V )
22		offval3 6305	. . 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 289	. 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 2290	1 |- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) = ( ( F |` D ) oF R ( G |` D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2202   _Vcvv 2803   i^i cin 3200   |-> cmpt 4155   dom cdm 4731   |` cres 4733   ` cfv 5333  (class class class)co 6028   oFcof 6242

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-of 6244

This theorem is referenced by: (None)