Theorem offres 5906

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 3221	. . . . . 6 |- ( ( dom F i^i dom G ) i^i D ) C_ D
2	1	sseli 3021	. . . . 5 |- ( x e. ( ( dom F i^i dom G ) i^i D ) -> x e. D )
3		fvres 5329	. . . . . 6 |- ( x e. D -> ( ( F |` D ) ` x ) = ( F ` x ) )
4		fvres 5329	. . . . . 6 |- ( x e. D -> ( ( G |` D ) ` x ) = ( G ` x ) )
5	3, 4	oveq12d 5670	. . . . 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 3924	. . 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 3217	. . . . 5 |- ( D i^i ( dom F i^i dom G ) ) = ( ( D i^i dom F ) i^i ( D i^i dom G ) )
9		incom 3192	. . . . 5 |- ( ( dom F i^i dom G ) i^i D ) = ( D i^i ( dom F i^i dom G ) )
10		dmres 4734	. . . . . 6 |- dom ( F |` D ) = ( D i^i dom F )
11		dmres 4734	. . . . . 6 |- dom ( G |` D ) = ( D i^i dom G )
12	10, 11	ineq12i 3199	. . . . 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 2119	. . . 4 |- ( dom ( F |` D ) i^i dom ( G |` D ) ) = ( ( dom F i^i dom G ) i^i D )
14		eqid 2088	. . . 4 |- ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) ) = ( ( ( F |` D ) ` x ) R ( ( G |` D ) ` x ) )
15	13, 14	mpteq12i 3926	. . 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 4761	. . 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 2119	. 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 5905	. . 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 4712	. 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 4752	. . 3 |- ( F e. V -> ( F |` D ) e. _V )
21		resexg 4752	. . 3 |- ( G e. W -> ( G |` D ) e. _V )
22		offval3 5905	. . 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 283	. 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 2146	1 |- ( ( F e. V /\ G e. W ) -> ( ( F oF R G ) |` D ) = ( ( F |` D ) oF R ( G |` D ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1289   e. wcel 1438   _Vcvv 2619   i^i cin 2998   |-> cmpt 3899   dom cdm 4438   |` cres 4440   ` cfv 5015  (class class class)co 5652   oFcof 5854

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-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-of 5856

This theorem is referenced by: (None)