Theorem ofvalg 6228

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 2230	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2230	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 6226	. . . 4 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
9	8	fveq1d 5629	. . 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 2229	. . 3 |- ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) )
12		fveq2 5627	. . . 4 |- ( x = X -> ( F ` x ) = ( F ` X ) )
13		fveq2 5627	. . . 4 |- ( x = X -> ( G ` x ) = ( G ` X ) )
14	12, 13	oveq12d 6019	. . 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 3424	. . . . . . . 8 |- ( A i^i B ) C_ A
17	5, 16	eqsstrri 3257	. . . . . . 7 |- S C_ A
18	17	sseli 3220	. . . . . 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 3425	. . . . . . . 8 |- ( A i^i B ) C_ B
22	5, 21	eqsstrri 3257	. . . . . . 7 |- S C_ B
23	22	sseli 3220	. . . . . 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 6019	. . . 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 2306	. . 3 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) e. U )
29	11, 14, 15, 28	fvmptd3 5728	. 2 |- ( ( ph /\ X e. S ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
30	10, 29, 26	3eqtrd 2266	1 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1395   e. wcel 2200   i^i cin 3196   |-> cmpt 4145   Fn wfn 5313   ` cfv 5318  (class class class)co 6001   oFcof 6216

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 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-setind 4629

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-ov 6004  df-oprab 6005  df-mpo 6006  df-of 6218

This theorem is referenced by:  offeq  6232  ofc1g  6240  ofc2g  6241  ofnegsub  9109  gsumfzmptfidmadd  13876  mplsubgfilemcl  14663  dvaddxxbr  15375  dvmulxxbr  15376  plyaddlem1  15421