Theorem ofvalg 5991

Description: Evaluate a function operation at a point. (Contributed by Mario Carneiro, 20-Jul-2014.) (Revised by Jim Kingdon, 22-Nov-2023.)

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 /\ X e. S ) -> ( C R D ) e. U )

Assertion

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

Proof of Theorem ofvalg

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 2140	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2140	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 5989	. . . 4 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
9	8	fveq1d 5423	. . 3 |- ( ph -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
10	9	adantr 274	. 2 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
11		eqid 2139	. . 3 |- ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) )
12		fveq2 5421	. . . 4 |- ( x = X -> ( F ` x ) = ( F ` X ) )
13		fveq2 5421	. . . 4 |- ( x = X -> ( G ` x ) = ( G ` X ) )
14	12, 13	oveq12d 5792	. . 3 |- ( x = X -> ( ( F ` x ) R ( G ` x ) ) = ( ( F ` X ) R ( G ` X ) ) )
15		simpr 109	. . 3 |- ( ( ph /\ X e. S ) -> X e. S )
16		inss1 3296	. . . . . . . 8 |- ( A i^i B ) C_ A
17	5, 16	eqsstrri 3130	. . . . . . 7 |- S C_ A
18	17	sseli 3093	. . . . . 6 |- ( X e. S -> X e. A )
19		ofval.6	. . . . . 6 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
20	18, 19	sylan2 284	. . . . 5 |- ( ( ph /\ X e. S ) -> ( F ` X ) = C )
21		inss2 3297	. . . . . . . 8 |- ( A i^i B ) C_ B
22	5, 21	eqsstrri 3130	. . . . . . 7 |- S C_ B
23	22	sseli 3093	. . . . . 6 |- ( X e. S -> X e. B )
24		ofval.7	. . . . . 6 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
25	23, 24	sylan2 284	. . . . 5 |- ( ( ph /\ X e. S ) -> ( G ` X ) = D )
26	20, 25	oveq12d 5792	. . . 4 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) = ( C R D ) )
27		ofval.8	. . . 4 |- ( ( ph /\ X e. S ) -> ( C R D ) e. U )
28	26, 27	eqeltrd 2216	. . 3 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) e. U )
29	11, 14, 15, 28	fvmptd3 5514	. 2 |- ( ( ph /\ X e. S ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
30	10, 29, 26	3eqtrd 2176	1 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   i^i cin 3070   |-> cmpt 3989   Fn wfn 5118   ` cfv 5123  (class class class)co 5774   oFcof 5980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982

This theorem is referenced by:  offeq  5995  dvaddxxbr  12834  dvmulxxbr  12835