Theorem ofres 6196

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 2208	. . 3 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2208	. . 3 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 6189	. 2 |- ( ph -> ( F oF R G ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
9		inss1 3401	. . . . 5 |- ( A i^i B ) C_ A
10	5, 9	eqsstrri 3234	. . . 4 |- C C_ A
11		fnssres 5408	. . . 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 3402	. . . . 5 |- ( A i^i B ) C_ B
14	5, 13	eqsstrri 3234	. . . 4 |- C C_ B
15		fnssres 5408	. . . 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 4199	. . . 4 |- ( ( C C_ A /\ A e. V ) -> C e. _V )
18	10, 3, 17	sylancr 414	. . 3 |- ( ph -> C e. _V )
19		inidm 3390	. . 3 |- ( C i^i C ) = C
20		fvres 5623	. . . 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 5623	. . . 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 6189	. 2 |- ( ph -> ( ( F |` C ) oF R ( G |` C ) ) = ( x e. C |-> ( ( F ` x ) R ( G ` x ) ) ) )
25	8, 24	eqtr4d 2243	1 |- ( ph -> ( F oF R G ) = ( ( F |` C ) oF R ( G |` C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   C_ wss 3174   |-> cmpt 4121   |` cres 4695   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   oFcof 6179

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181

This theorem is referenced by: (None)