Theorem ofres 5907

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 2096	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2096	. . 3 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 5901	. 2 |- ( ph -> ( F oF R G ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
9		inss1 3235	. . . . 5 |- ( A i^i B ) C_ A
10	5, 9	eqsstr3i 3072	. . . 4 |- C C_ A
11		fnssres 5161	. . . 4 |- ( ( F Fn A /\ C C_ A ) -> ( F |` C ) Fn C )
12	1, 10, 11	sylancl 405	. . 3 |- ( ph -> ( F |` C ) Fn C )
13		inss2 3236	. . . . 5 |- ( A i^i B ) C_ B
14	5, 13	eqsstr3i 3072	. . . 4 |- C C_ B
15		fnssres 5161	. . . 4 |- ( ( G Fn B /\ C C_ B ) -> ( G |` C ) Fn C )
16	2, 14, 15	sylancl 405	. . 3 |- ( ph -> ( G |` C ) Fn C )
17		ssexg 3999	. . . 4 |- ( ( C C_ A /\ A e. V ) -> C e. _V )
18	10, 3, 17	sylancr 406	. . 3 |- ( ph -> C e. _V )
19		inidm 3224	. . 3 |- ( C i^i C ) = C
20		fvres 5364	. . . 4 |- ( x e. C -> ( ( F |` C ) ` x ) = ( F ` x ) )
21	20	adantl 272	. . 3 |- ( ( ph /\ x e. C ) -> ( ( F |` C ) ` x ) = ( F ` x ) )
22		fvres 5364	. . . 4 |- ( x e. C -> ( ( G |` C ) ` x ) = ( G ` x ) )
23	22	adantl 272	. . 3 |- ( ( ph /\ x e. C ) -> ( ( G |` C ) ` x ) = ( G ` x ) )
24	12, 16, 18, 18, 19, 21, 23	offval 5901	. 2 |- ( ph -> ( ( F |` C ) oF R ( G |` C ) ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
25	8, 24	eqtr4d 2130	1 |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1296   e. wcel 1445   _Vcvv 2633   i^i cin 3012   C_ wss 3013   |-> cmpt 3921   |` cres 4469   Fn wfn 5044   ` cfv 5049  (class class class)co 5690   oFcof 5892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-of 5894

This theorem is referenced by: (None)