Theorem offres 6372

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 3452	. . . . . 6 |- ( ( dom F i^i dom G ) i^i D ) C_ D
2	1	sseli 3244	. . . . 5 |- ( x e. ( ( dom F i^i dom G ) i^i D ) -> x e. D )
3		fvres 5722	. . . . . 6 |- ( x e. D -> ( ( F |` D ) ` x ) = ( F ` x ) )
4		fvres 5722	. . . . . 6 |- ( x e. D -> ( ( G |` D ) ` x ) = ( G ` x ) )
5	3, 4	oveq12d 6106	. . . . 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 4219	. . 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 3448	. . . . 5 |- ( D i^i ( dom F i^i dom G ) ) = ( ( D i^i dom F ) i^i ( D i^i dom G ) )
9		incom 3421	. . . . 5 |- ( ( dom F i^i dom G ) i^i D ) = ( D i^i ( dom F i^i dom G ) )
10		dmres 5086	. . . . . 6 |- dom ( F |` D ) = ( D i^i dom F )
11		dmres 5086	. . . . . 6 |- dom ( G |` D ) = ( D i^i dom G )
12	10, 11	ineq12i 3430	. . . . 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 2270	. . . 4 |- ( dom ( F |` D ) i^i dom ( G |` D ) ) = ( ( dom F i^i dom G ) i^i D )
14		eqid 2238	. . . 4 |- ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) = ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) )
15	13, 14	mpteq12i 4221	. . 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 5115	. . 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 2270	. 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 6371	. . 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 5064	. 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 5105	. . 3 |- ( F e. V -> ( F |` D ) e. _V )
21		resexg 5105	. . 3 |- ( G e. W -> ( G |` D ) e. _V )
22		offval3 6371	. . 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 2297	1 |- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) = ( ( F |` D ) oF R ( G |` D ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   _Vcvv 2821   i^i cin 3219   |-> cmpt 4193   dom cdm 4776   |` cres 4778   ` cfv 5380  (class class class)co 6088   oFcof 6303

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-of 6305

This theorem is used by: (None)