Theorem ofres 6004

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 2141	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2141	. . 3 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 5997	. 2 |- ( ph -> ( F oF R G ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
9		inss1 3301	. . . . 5 |- ( A i^i B ) C_ A
10	5, 9	eqsstrri 3135	. . . 4 |- C C_ A
11		fnssres 5244	. . . 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 3302	. . . . 5 |- ( A i^i B ) C_ B
14	5, 13	eqsstrri 3135	. . . 4 |- C C_ B
15		fnssres 5244	. . . 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 4075	. . . 4 |- ( ( C C_ A /\ A e. V ) -> C e. _V )
18	10, 3, 17	sylancr 411	. . 3 |- ( ph -> C e. _V )
19		inidm 3290	. . 3 |- ( C i^i C ) = C
20		fvres 5453	. . . 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 5453	. . . 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 5997	. 2 |- ( ph -> ( ( F |` C ) oF R ( G |` C ) ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
25	8, 24	eqtr4d 2176	1 |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   _Vcvv 2689   i^i cin 3075   C_ wss 3076   |-> cmpt 3997   |` cres 4549   Fn wfn 5126   ` cfv 5131  (class class class)co 5782   oFcof 5988

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-id 4223  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-of 5990

This theorem is referenced by: (None)