Theorem ofval 6906

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 )

Assertion

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

Proof of Theorem ofval

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 2623	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
7		eqidd 2623	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = ( G ` x ) )
8	1, 2, 3, 4, 5, 6, 7	offval 6904	. . . 4 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
9	8	fveq1d 6193	. . 3 |- ( ph -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
10	9	adantr 481	. 2 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) )
11		fveq2 6191	. . . . 5 |- ( x = X -> ( F ` x ) = ( F ` X ) )
12		fveq2 6191	. . . . 5 |- ( x = X -> ( G ` x ) = ( G ` X ) )
13	11, 12	oveq12d 6668	. . . 4 |- ( x = X -> ( ( F ` x ) R ( G ` x ) ) = ( ( F ` X ) R ( G ` X ) ) )
14		eqid 2622	. . . 4 |- ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) )
15		ovex 6678	. . . 4 |- ( ( F ` X ) R ( G ` X ) ) e. _V
16	13, 14, 15	fvmpt 6282	. . 3 |- ( X e. S -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
17	16	adantl 482	. 2 |- ( ( ph /\ X e. S ) -> ( ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) ` X ) = ( ( F ` X ) R ( G ` X ) ) )
18		inss1 3833	. . . . . 6 |- ( A i^i B ) C_ A
19	5, 18	eqsstr3i 3636	. . . . 5 |- S C_ A
20	19	sseli 3599	. . . 4 |- ( X e. S -> X e. A )
21		ofval.6	. . . 4 |- ( ( ph /\ X e. A ) -> ( F ` X ) = C )
22	20, 21	sylan2 491	. . 3 |- ( ( ph /\ X e. S ) -> ( F ` X ) = C )
23		inss2 3834	. . . . . 6 |- ( A i^i B ) C_ B
24	5, 23	eqsstr3i 3636	. . . . 5 |- S C_ B
25	24	sseli 3599	. . . 4 |- ( X e. S -> X e. B )
26		ofval.7	. . . 4 |- ( ( ph /\ X e. B ) -> ( G ` X ) = D )
27	25, 26	sylan2 491	. . 3 |- ( ( ph /\ X e. S ) -> ( G ` X ) = D )
28	22, 27	oveq12d 6668	. 2 |- ( ( ph /\ X e. S ) -> ( ( F ` X ) R ( G ` X ) ) = ( C R D ) )
29	10, 17, 28	3eqtrd 2660	1 |- ( ( ph /\ X e. S ) -> ( ( F oF R G ) ` X ) = ( C R D ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   |-> cmpt 4729   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   oFcof 6895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897

This theorem is referenced by:  fnfvof  6911  offveq  6918  ofc1  6920  ofc2  6921  suppofss1d  7332  suppofss2d  7333  ofsubeq0  11017  ofnegsub  11018  ofsubge0  11019  seqof  12858  o1of2  14343  gsumzaddlem  18321  psrbagcon  19371  psrbagconf1o  19374  psrdi  19406  psrdir  19407  mplsubglem  19434  matplusgcell  20239  matsubgcell  20240  rrxcph  23180  mbfaddlem  23427  i1faddlem  23460  i1fmullem  23461  itg1lea  23479  mbfi1flimlem  23489  itg2split  23516  itg2monolem1  23517  itg2addlem  23525  dvaddbr  23701  dvmulbr  23702  plyaddlem1  23969  coeeulem  23980  coeaddlem  24005  dgradd2  24024  dgrcolem2  24030  ofmulrt  24037  plydivlem3  24050  plydivlem4  24051  plydiveu  24053  plyrem  24060  vieta1lem2  24066  elqaalem3  24076  qaa  24078  basellem7  24813  basellem9  24815  circlemethhgt  30721  poimirlem1  33410  poimirlem2  33411  poimirlem6  33415  poimirlem7  33416  poimirlem10  33419  poimirlem11  33420  poimirlem12  33421  poimirlem17  33426  poimirlem20  33429  poimirlem23  33432  poimirlem29  33438  poimirlem31  33440  poimirlem32  33441  broucube  33443  itg2addnclem3  33463  itg2addnc  33464  ftc1anclem5  33489  lfladdcl  34358  ldualvaddval  34418  dgrsub2  37705  mpaaeu  37720  caofcan  38522  ofmul12  38524  ofdivrec  38525  ofdivcan4  38526  ofdivdiv2  38527  binomcxplemrat  38549  binomcxplemnotnn0  38555  mndpsuppss  42152  amgmwlem  42548