Theorem ofvalg 6276

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 2233	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2233	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 6274	. . . 4 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
9	8	fveq1d 5672	. . 3 |- ( ph -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
10	9	adantr 276	. 2 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
11		eqid 2232	. . 3 |- ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) )
12		fveq2 5670	. . . 4 |- ( x = X -> ( F ` x ) = ( F ` X ) )
13		fveq2 5670	. . . 4 |- ( x = X -> ( G ` x ) = ( G ` X ) )
14	12, 13	oveq12d 6068	. . 3 |- ( x = X -> ( ( F ` x ) R ( G ` x ) ) = ( ( F ` X ) R ( G ` X ) ) )
15		simpr 110	. . 3 |- ( ( ph /\ X e. S ) -> X e. S )
16		inss1 3441	. . . . . . . 8 |- ( A i^i B ) C_ A
17	5, 16	eqsstrri 3271	. . . . . . 7 |- S C_ A
18	17	sseli 3234	. . . . . 6 |- ( X e. S -> X e. A )
19		ofval.6	. . . . . 6 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
20	18, 19	sylan2 286	. . . . 5 |- ( ( ph /\ X e. S ) -> ( F ` X ) = C )
21		inss2 3442	. . . . . . . 8 |- ( A i^i B ) C_ B
22	5, 21	eqsstrri 3271	. . . . . . 7 |- S C_ B
23	22	sseli 3234	. . . . . 6 |- ( X e. S -> X e. B )
24		ofval.7	. . . . . 6 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
25	23, 24	sylan2 286	. . . . 5 |- ( ( ph /\ X e. S ) -> ( G ` X ) = D )
26	20, 25	oveq12d 6068	. . . 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 2309	. . 3 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) e. U )
29	11, 14, 15, 28	fvmptd3 5771	. 2 |- ( ( ph /\ X e. S ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
30	10, 29, 26	3eqtrd 2269	1 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2203   i^i cin 3210   |-> cmpt 4171   Fn wfn 5347   ` cfv 5352  (class class class)co 6050   oFcof 6264

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-ov 6053  df-oprab 6054  df-mpo 6055  df-of 6266

This theorem is referenced by:  offeq  6280  ofc1g  6288  ofc2g  6289  suppofss1dcl  6464  suppofss2dcl  6465  ofnegsub  9236  gsumfzmptfidmadd  14056  psrbagcon  14826  psrbagconf1o  14828  mplsubgfilemcl  14854  dvaddxxbr  15566  dvmulxxbr  15567  plyaddlem1  15612