Theorem ofres 6064

Description: Restrict the operands of a function operation to the same domain as that of the operation itself. (Contributed by Mario Carneiro, 15-Sep-2014.)

Hypotheses

Ref	Expression
ofres.1	|- ( ph -> F Fn A )
ofres.2	|- ( ph -> G Fn B )
ofres.3	|- ( ph -> A e. V )
ofres.4	|- ( ph -> B e. W )
ofres.5	|- ( A i^i B ) = C

Assertion

Ref	Expression
ofres	|- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) )

Proof of Theorem ofres

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofres.1	. . 3 |- ( ph -> F Fn A )
2		ofres.2	. . 3 |- ( ph -> G Fn B )
3		ofres.3	. . 3 |- ( ph -> A e. V )
4		ofres.4	. . 3 |- ( ph -> B e. W )
5		ofres.5	. . 3 |- ( A i^i B ) = C
6		eqidd 2166	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2166	. . 3 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 6057	. 2 |- ( ph -> ( F oF R G ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
9		inss1 3342	. . . . 5 |- ( A i^i B ) C_ A
10	5, 9	eqsstrri 3175	. . . 4 |- C C_ A
11		fnssres 5301	. . . 4 |- ( ( F Fn A /\ C C_ A ) -> ( F |` C ) Fn C )
12	1, 10, 11	sylancl 410	. . 3 |- ( ph -> ( F |` C ) Fn C )
13		inss2 3343	. . . . 5 |- ( A i^i B ) C_ B
14	5, 13	eqsstrri 3175	. . . 4 |- C C_ B
15		fnssres 5301	. . . 4 |- ( ( G Fn B /\ C C_ B ) -> ( G |` C ) Fn C )
16	2, 14, 15	sylancl 410	. . 3 |- ( ph -> ( G |` C ) Fn C )
17		ssexg 4121	. . . 4 |- ( ( C C_ A /\ A e. V ) -> C e. _V )
18	10, 3, 17	sylancr 411	. . 3 |- ( ph -> C e. _V )
19		inidm 3331	. . 3 |- ( C i^i C ) = C
20		fvres 5510	. . . 4 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
21	20	adantl 275	. . 3 |- ( ( ph /\ x e. C ) -> ( ( F |` C ) ` x ) = ( F ` x ) )
22		fvres 5510	. . . 4 |- ( x e. C -> ( ( G |` C ) ` x ) = ( G ` x ) )
23	22	adantl 275	. . 3 |- ( ( ph /\ x e. C ) -> ( ( G |` C ) ` x ) = ( G ` x ) )
24	12, 16, 18, 18, 19, 21, 23	offval 6057	. 2 |- ( ph -> ( ( F |` C ) oF R ( G |` C ) ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
25	8, 24	eqtr4d 2201	1 |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   C_ wss 3116   |-> cmpt 4043   |` cres 4606   Fn wfn 5183   ` cfv 5188  (class class class)co 5842   oFcof 6048

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050

This theorem is referenced by: (None)