Theorem fnofval 5752

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 2083	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2083	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 5750	. . . 4 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
9	8	fveq1d 5211	. . 3 |- ( ph -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
10	9	adantr 270	. 2 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
11		simpr 108	. . 3 |- ( ( ph /\ X e. S ) -> X e. S )
12		ofval.8	. . . . 5 |- ( ph -> R Fn ( U X. V ) )
13	12	adantr 270	. . . 4 |- ( ( ph /\ X e. S ) -> R Fn ( U X. V ) )
14		ofval.9	. . . . . 6 |- ( ph -> C e. U )
15	14	adantr 270	. . . . 5 |- ( ( ph /\ X e. S ) -> C e. U )
16		inss1 3193	. . . . . . . . 9 |- ( A i^i B ) C_ A
17	5, 16	eqsstr3i 3031	. . . . . . . 8 |- S C_ A
18	17	sseli 2996	. . . . . . 7 |- ( X e. S -> X e. A )
19		ofval.6	. . . . . . 7 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
20	18, 19	sylan2 280	. . . . . 6 |- ( ( ph /\ X e. S ) -> ( F ` X ) = C )
21	20	eleq1d 2148	. . . . 5 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) e. U <-> C e. U ) )
22	15, 21	mpbird 165	. . . 4 |- ( ( ph /\ X e. S ) -> ( F ` X ) e. U )
23		ofval.10	. . . . . 6 |- ( ph -> D e. V )
24	23	adantr 270	. . . . 5 |- ( ( ph /\ X e. S ) -> D e. V )
25		inss2 3194	. . . . . . . . 9 |- ( A i^i B ) C_ B
26	5, 25	eqsstr3i 3031	. . . . . . . 8 |- S C_ B
27	26	sseli 2996	. . . . . . 7 |- ( X e. S -> X e. B )
28		ofval.7	. . . . . . 7 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
29	27, 28	sylan2 280	. . . . . 6 |- ( ( ph /\ X e. S ) -> ( G ` X ) = D )
30	29	eleq1d 2148	. . . . 5 |- ( ( ph /\ X e. S ) -> ( ( G ` X ) e. V <-> D e. V ) )
31	24, 30	mpbird 165	. . . 4 |- ( ( ph /\ X e. S ) -> ( G ` X ) e. V )
32		fnovex 5569	. . . 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 1170	. . 3 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) e. _V )
34		fveq2 5209	. . . . 5 |- ( x = X -> ( F ` x ) = ( F ` X ) )
35		fveq2 5209	. . . . 5 |- ( x = X -> ( G ` x ) = ( G ` X ) )
36	34, 35	oveq12d 5561	. . . 4 |- ( x = X -> ( ( F ` x ) R ( G ` x ) ) = ( ( F ` X ) R ( G ` X ) ) )
37		eqid 2082	. . . 4 |- ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) )
38	36, 37	fvmptg 5280	. . 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 403	. 2 |- ( ( ph /\ X e. S ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
40	20, 29	oveq12d 5561	. 2 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) = ( C R D ) )
41	10, 39, 40	3eqtrd 2118	1 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   _Vcvv 2602   i^i cin 2973   |-> cmpt 3847   X. cxp 4369   Fn wfn 4927   ` cfv 4932  (class class class)co 5543   oFcof 5741

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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-of 5743

This theorem is referenced by: (None)