Theorem ofres 6087

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 2176	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2176	. . 3 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 6080	. 2 |- ( ph -> ( F oF R G ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
9		inss1 3353	. . . . 5 |- ( A i^i B ) C_ A
10	5, 9	eqsstrri 3186	. . . 4 |- C C_ A
11		fnssres 5321	. . . 4 |- ( ( F Fn A /\ C C_ A ) -> ( F |` C ) Fn C )
12	1, 10, 11	sylancl 413	. . 3 |- ( ph -> ( F |` C ) Fn C )
13		inss2 3354	. . . . 5 |- ( A i^i B ) C_ B
14	5, 13	eqsstrri 3186	. . . 4 |- C C_ B
15		fnssres 5321	. . . 4 |- ( ( G Fn B /\ C C_ B ) -> ( G |` C ) Fn C )
16	2, 14, 15	sylancl 413	. . 3 |- ( ph -> ( G |` C ) Fn C )
17		ssexg 4137	. . . 4 |- ( ( C C_ A /\ A e. V ) -> C e. _V )
18	10, 3, 17	sylancr 414	. . 3 |- ( ph -> C e. _V )
19		inidm 3342	. . 3 |- ( C i^i C ) = C
20		fvres 5531	. . . 4 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
21	20	adantl 277	. . 3 |- ( ( ph /\ x e. C ) -> ( ( F |` C ) ` x ) = ( F ` x ) )
22		fvres 5531	. . . 4 |- ( x e. C -> ( ( G |` C ) ` x ) = ( G ` x ) )
23	22	adantl 277	. . 3 |- ( ( ph /\ x e. C ) -> ( ( G |` C ) ` x ) = ( G ` x ) )
24	12, 16, 18, 18, 19, 21, 23	offval 6080	. 2 |- ( ph -> ( ( F |` C ) oF R ( G |` C ) ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
25	8, 24	eqtr4d 2211	1 |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   _Vcvv 2735   i^i cin 3126   C_ wss 3127   |-> cmpt 4059   |` cres 4622   Fn wfn 5203   ` cfv 5208  (class class class)co 5865   oFcof 6071

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-coll 4113  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-csb 3056  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-iun 3884  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-res 4632  df-ima 4633  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-ov 5868  df-oprab 5869  df-mpo 5870  df-of 6073

This theorem is referenced by: (None)