Theorem ofco 5749

Description: The composition of a function operation with another function. (Contributed by Mario Carneiro, 19-Dec-2014.)

Hypotheses

Ref	Expression
ofco.1	|- ( ph -> F Fn A )
ofco.2	|- ( ph -> G Fn B )
ofco.3	|- ( ph -> H : D --> C )
ofco.4	|- ( ph -> A e. V )
ofco.5	|- ( ph -> B e. W )
ofco.6	|- ( ph -> D e. X )
ofco.7	|- ( A i^i B ) = C

Assertion

Ref	Expression
ofco	|- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )

Proof of Theorem ofco

Dummy variables y x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ofco.3	. . . 4 |- ( ph -> H : D --> C )
2	1	ffvelrnda 5323	. . 3 |- ( ( ph /\ x e. D ) -> ( H ` x ) e. C )
3	1	feqmptd 5247	. . 3 |- ( ph -> H = ( x e. D |-> ( H ` x ) ) )
4		ofco.1	. . . 4 |- ( ph -> F Fn A )
5		ofco.2	. . . 4 |- ( ph -> G Fn B )
6		ofco.4	. . . 4 |- ( ph -> A e. V )
7		ofco.5	. . . 4 |- ( ph -> B e. W )
8		ofco.7	. . . 4 |- ( A i^i B ) = C
9		eqidd 2082	. . . 4 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( F ` y ) )
10		eqidd 2082	. . . 4 |- ( ( ph /\ y e. B ) -> ( G ` y ) = ( G ` y ) )
11	4, 5, 6, 7, 8, 9, 10	offval 5739	. . 3 |- ( ph -> ( F oF R G ) = ( y e. C |-> ( ( F ` y ) R ( G ` y ) ) ) )
12		fveq2 5198	. . . 4 |- ( y = ( H ` x ) -> ( F ` y ) = ( F ` ( H ` x ) ) )
13		fveq2 5198	. . . 4 |- ( y = ( H ` x ) -> ( G ` y ) = ( G ` ( H ` x ) ) )
14	12, 13	oveq12d 5550	. . 3 |- ( y = ( H ` x ) -> ( ( F ` y ) R ( G ` y ) ) = ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) )
15	2, 3, 11, 14	fmptco 5351	. 2 |- ( ph -> ( ( F oF R G ) o. H ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) )
16		inss1 3186	. . . . . 6 |- ( A i^i B ) C_ A
17	8, 16	eqsstr3i 3030	. . . . 5 |- C C_ A
18		fss 5074	. . . . 5 |- ( ( H : D --> C /\ C C_ A ) -> H : D --> A )
19	1, 17, 18	sylancl 404	. . . 4 |- ( ph -> H : D --> A )
20		fnfco 5085	. . . 4 |- ( ( F Fn A /\ H : D --> A ) -> ( F o. H ) Fn D )
21	4, 19, 20	syl2anc 403	. . 3 |- ( ph -> ( F o. H ) Fn D )
22		inss2 3187	. . . . . 6 |- ( A i^i B ) C_ B
23	8, 22	eqsstr3i 3030	. . . . 5 |- C C_ B
24		fss 5074	. . . . 5 |- ( ( H : D --> C /\ C C_ B ) -> H : D --> B )
25	1, 23, 24	sylancl 404	. . . 4 |- ( ph -> H : D --> B )
26		fnfco 5085	. . . 4 |- ( ( G Fn B /\ H : D --> B ) -> ( G o. H ) Fn D )
27	5, 25, 26	syl2anc 403	. . 3 |- ( ph -> ( G o. H ) Fn D )
28		ofco.6	. . 3 |- ( ph -> D e. X )
29		inidm 3175	. . 3 |- ( D i^i D ) = D
30		ffn 5066	. . . . 5 |- ( H : D --> C -> H Fn D )
31	1, 30	syl 14	. . . 4 |- ( ph -> H Fn D )
32		fvco2 5263	. . . 4 |- ( ( H Fn D /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
33	31, 32	sylan 277	. . 3 |- ( ( ph /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
34		fvco2 5263	. . . 4 |- ( ( H Fn D /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) )
35	31, 34	sylan 277	. . 3 |- ( ( ph /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) )
36	21, 27, 28, 28, 29, 33, 35	offval 5739	. 2 |- ( ph -> ( ( F o. H ) oF R ( G o. H ) ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) )
37	15, 36	eqtr4d 2116	1 |- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   i^i cin 2972   C_ wss 2973   |-> cmpt 3839   o. ccom 4367   Fn wfn 4917   -->wf 4918   ` cfv 4922  (class class class)co 5532   oFcof 5730

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-of 5732

This theorem is referenced by: (None)