Theorem offres 6296

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 3428	. . . . . 6 |- ( ( dom F i^i dom G ) i^i D ) C_ D
2	1	sseli 3223	. . . . 5 |- ( x e. ( ( dom F i^i dom G ) i^i D ) -> x e. D )
3		fvres 5663	. . . . . 6 |- ( x e. D -> ( ( F |` D ) ` x ) = ( F ` x ) )
4		fvres 5663	. . . . . 6 |- ( x e. D -> ( ( G |` D ) ` x ) = ( G ` x ) )
5	3, 4	oveq12d 6035	. . . . 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 4175	. . 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 3424	. . . . 5 |- ( D i^i ( dom F i^i dom G ) ) = ( ( D i^i dom F ) i^i ( D i^i dom G ) )
9		incom 3399	. . . . 5 |- ( ( dom F i^i dom G ) i^i D ) = ( D i^i ( dom F i^i dom G ) )
10		dmres 5034	. . . . . 6 |- dom ( F |` D ) = ( D i^i dom F )
11		dmres 5034	. . . . . 6 |- dom ( G |` D ) = ( D i^i dom G )
12	10, 11	ineq12i 3406	. . . . 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 4177	. . 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 5062	. . 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 6295	. . 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 5012	. 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 5053	. . 3 |- ( F e. V -> ( F |` D ) e. _V )
21		resexg 5053	. . 3 |- ( G e. W -> ( G |` D ) e. _V )
22		offval3 6295	. . 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 1397   e. wcel 2202   _Vcvv 2802   i^i cin 3199   |-> cmpt 4150   dom cdm 4725   |` cres 4727   ` cfv 5326  (class class class)co 6017   oFcof 6232

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-of 6234

This theorem is referenced by: (None)