Theorem fnofval 5721

Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.)

Hypotheses

Ref	Expression
offval.1	|- ( ph -> F Fn A )
offval.2	|- ( ph -> G Fn B )
offval.3	|- ( ph -> A e. V )
offval.4	|- ( ph -> B e. W )
offval.5	|- ( A i^i B ) = S
ofval.6	|- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
ofval.7	|- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
ofval.8	|- ( ph -> R Fn ( U X. V ) )
ofval.9	|- ( ph -> C e. U )
ofval.10	|- ( ph -> D e. V )

Assertion

Ref	Expression
fnofval	|- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Proof of Theorem fnofval

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . . 5 |- ( ph -> F Fn A )
2		offval.2	. . . . 5 |- ( ph -> G Fn B )
3		offval.3	. . . . 5 |- ( ph -> A e. V )
4		offval.4	. . . . 5 |- ( ph -> B e. W )
5		offval.5	. . . . 5 |- ( A i^i B ) = S
6		eqidd 2041	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2041	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 5719	. . . 4 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
9	8	fveq1d 5180	. . 3 |- ( ph -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
10	9	adantr 261	. 2 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
11		simpr 103	. . 3 |- ( ( ph /\ X e. S ) -> X e. S )
12		ofval.8	. . . . 5 |- ( ph -> R Fn ( U X. V ) )
13	12	adantr 261	. . . 4 |- ( ( ph /\ X e. S ) -> R Fn ( U X. V ) )
14		ofval.9	. . . . . 6 |- ( ph -> C e. U )
15	14	adantr 261	. . . . 5 |- ( ( ph /\ X e. S ) -> C e. U )
16		inss1 3157	. . . . . . . . 9 |- ( A i^i B ) C_ A
17	5, 16	eqsstr3i 2976	. . . . . . . 8 |- S C_ A
18	17	sseli 2941	. . . . . . 7 |- ( X e. S -> X e. A )
19		ofval.6	. . . . . . 7 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
20	18, 19	sylan2 270	. . . . . 6 |- ( ( ph /\ X e. S ) -> ( F ` X ) = C )
21	20	eleq1d 2106	. . . . 5 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) e. U <-> C e. U ) )
22	15, 21	mpbird 156	. . . 4 |- ( ( ph /\ X e. S ) -> ( F ` X ) e. U )
23		ofval.10	. . . . . 6 |- ( ph -> D e. V )
24	23	adantr 261	. . . . 5 |- ( ( ph /\ X e. S ) -> D e. V )
25		inss2 3158	. . . . . . . . 9 |- ( A i^i B ) C_ B
26	5, 25	eqsstr3i 2976	. . . . . . . 8 |- S C_ B
27	26	sseli 2941	. . . . . . 7 |- ( X e. S -> X e. B )
28		ofval.7	. . . . . . 7 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
29	27, 28	sylan2 270	. . . . . 6 |- ( ( ph /\ X e. S ) -> ( G ` X ) = D )
30	29	eleq1d 2106	. . . . 5 |- ( ( ph /\ X e. S ) -> ( ( G ` X ) e. V <-> D e. V ) )
31	24, 30	mpbird 156	. . . 4 |- ( ( ph /\ X e. S ) -> ( G ` X ) e. V )
32		fnovex 5538	. . . 4 |- ( ( R Fn ( U X. V ) /\ ( F ` X ) e. U /\ ( G ` X ) e. V ) -> ( ( F ` X ) R ( G ` X ) ) e. _V )
33	13, 22, 31, 32	syl3anc 1135	. . 3 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) e. _V )
34		fveq2 5178	. . . . 5 |- ( x = X -> ( F ` x ) = ( F ` X ) )
35		fveq2 5178	. . . . 5 |- ( x = X -> ( G ` x ) = ( G ` X ) )
36	34, 35	oveq12d 5530	. . . 4 |- ( x = X -> ( ( F ` x ) R ( G ` x ) ) = ( ( F ` X ) R ( G ` X ) ) )
37		eqid 2040	. . . 4 |- ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) )
38	36, 37	fvmptg 5248	. . 3 |- ( ( X e. S /\ ( ( F ` X ) R ( G ` X ) ) e. _V ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
39	11, 33, 38	syl2anc 391	. 2 |- ( ( ph /\ X e. S ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
40	20, 29	oveq12d 5530	. 2 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) = ( C R D ) )
41	10, 39, 40	3eqtrd 2076	1 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   _Vcvv 2557   i^i cin 2916   |-> cmpt 3818   X. cxp 4343   Fn wfn 4897   ` cfv 4902  (class class class)co 5512   oFcof 5710

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-of 5712

This theorem is referenced by: (None)