Theorem ofco 6249

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	ffvelcdmda 5778	. . 3 |- ( ( ph /\ x e. D ) -> ( H ` x ) e. C )
3	1	feqmptd 5695	. . 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 2230	. . . 4 |- ( ( ph /\ y e. A ) -> ( F ` y ) = ( F ` y ) )
10		eqidd 2230	. . . 4 |- ( ( ph /\ y e. B ) -> ( G ` y ) = ( G ` y ) )
11	4, 5, 6, 7, 8, 9, 10	offval 6238	. . 3 |- ( ph -> ( F oF R G ) = ( y e. C |-> ( ( F ` y ) R ( G ` y ) ) ) )
12		fveq2 5635	. . . 4 |- ( y = ( H ` x ) -> ( F ` y ) = ( F ` ( H ` x ) ) )
13		fveq2 5635	. . . 4 |- ( y = ( H ` x ) -> ( G ` y ) = ( G ` ( H ` x ) ) )
14	12, 13	oveq12d 6031	. . 3 |- ( y = ( H ` x ) -> ( ( F ` y ) R ( G ` y ) ) = ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) )
15	2, 3, 11, 14	fmptco 5809	. 2 |- ( ph -> ( ( F oF R G ) o. H ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) )
16		inss1 3425	. . . . . 6 |- ( A i^i B ) C_ A
17	8, 16	eqsstrri 3258	. . . . 5 |- C C_ A
18		fss 5491	. . . . 5 |- ( ( H : D --> C /\ C C_ A ) -> H : D --> A )
19	1, 17, 18	sylancl 413	. . . 4 |- ( ph -> H : D --> A )
20		fnfco 5508	. . . 4 |- ( ( F Fn A /\ H : D --> A ) -> ( F o. H ) Fn D )
21	4, 19, 20	syl2anc 411	. . 3 |- ( ph -> ( F o. H ) Fn D )
22		inss2 3426	. . . . . 6 |- ( A i^i B ) C_ B
23	8, 22	eqsstrri 3258	. . . . 5 |- C C_ B
24		fss 5491	. . . . 5 |- ( ( H : D --> C /\ C C_ B ) -> H : D --> B )
25	1, 23, 24	sylancl 413	. . . 4 |- ( ph -> H : D --> B )
26		fnfco 5508	. . . 4 |- ( ( G Fn B /\ H : D --> B ) -> ( G o. H ) Fn D )
27	5, 25, 26	syl2anc 411	. . 3 |- ( ph -> ( G o. H ) Fn D )
28		ofco.6	. . 3 |- ( ph -> D e. X )
29		inidm 3414	. . 3 |- ( D i^i D ) = D
30		ffn 5479	. . . . 5 |- ( H : D --> C -> H Fn D )
31	1, 30	syl 14	. . . 4 |- ( ph -> H Fn D )
32		fvco2 5711	. . . 4 |- ( ( H Fn D /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
33	31, 32	sylan 283	. . 3 |- ( ( ph /\ x e. D ) -> ( ( F o. H ) ` x ) = ( F ` ( H ` x ) ) )
34		fvco2 5711	. . . 4 |- ( ( H Fn D /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) )
35	31, 34	sylan 283	. . 3 |- ( ( ph /\ x e. D ) -> ( ( G o. H ) ` x ) = ( G ` ( H ` x ) ) )
36	21, 27, 28, 28, 29, 33, 35	offval 6238	. 2 |- ( ph -> ( ( F o. H ) oF R ( G o. H ) ) = ( x e. D |-> ( ( F ` ( H ` x ) ) R ( G ` ( H ` x ) ) ) ) )
37	15, 36	eqtr4d 2265	1 |- ( ph -> ( ( F oF R G ) o. H ) = ( ( F o. H ) oF R ( G o. H ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   i^i cin 3197   C_ wss 3198   |-> cmpt 4148   o. ccom 4727   Fn wfn 5319   -->wf 5320   ` cfv 5324  (class class class)co 6013   oFcof 6228

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-of 6230

This theorem is referenced by: (None)