Theorem offval 6904

Description: Value of an operation applied to two functions. (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
offval.6	|- ( ( ph /\ x e. A ) -> ( F ` x ) = C )
offval.7	|- ( ( ph /\ x e. B ) -> ( G ` x ) = D )

Assertion

Ref	Expression
offval	|- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) )

Distinct variable groups:   x, A   x, F   x, G   ph, x   x, S   x, R

Allowed substitution hints:   B( x)   C( x)   D( x)   V( x)   W( x)

Proof of Theorem offval

Dummy variables f g are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		offval.1	. . . 4 |- ( ph -> F Fn A )
2		offval.3	. . . 4 |- ( ph -> A e. V )
3		fnex 6481	. . . 4 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
4	1, 2, 3	syl2anc 693	. . 3 |- ( ph -> F e. _V )
5		offval.2	. . . 4 |- ( ph -> G Fn B )
6		offval.4	. . . 4 |- ( ph -> B e. W )
7		fnex 6481	. . . 4 |- ( ( G Fn B /\ B e. W ) -> G e. _V )
8	5, 6, 7	syl2anc 693	. . 3 |- ( ph -> G e. _V )
9		fndm 5990	. . . . . . . 8 |- ( F Fn A -> dom F = A )
10	1, 9	syl 17	. . . . . . 7 |- ( ph -> dom F = A )
11		fndm 5990	. . . . . . . 8 |- ( G Fn B -> dom G = B )
12	5, 11	syl 17	. . . . . . 7 |- ( ph -> dom G = B )
13	10, 12	ineq12d 3815	. . . . . 6 |- ( ph -> ( dom F i^i dom G ) = ( A i^i B ) )
14		offval.5	. . . . . 6 |- ( A i^i B ) = S
15	13, 14	syl6eq 2672	. . . . 5 |- ( ph -> ( dom F i^i dom G ) = S )
16	15	mpteq1d 4738	. . . 4 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) )
17		inex1g 4801	. . . . . 6 |- ( A e. V -> ( A i^i B ) e. _V )
18	14, 17	syl5eqelr 2706	. . . . 5 |- ( A e. V -> S e. _V )
19		mptexg 6484	. . . . 5 |- ( S e. _V -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
20	2, 18, 19	3syl 18	. . . 4 |- ( ph -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
21	16, 20	eqeltrd 2701	. . 3 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
22		dmeq 5324	. . . . . 6 |- ( f = F -> dom f = dom F )
23		dmeq 5324	. . . . . 6 |- ( g = G -> dom g = dom G )
24	22, 23	ineqan12d 3816	. . . . 5 |- ( ( f = F /\ g = G ) -> ( dom f i^i dom g ) = ( dom F i^i dom G ) )
25		fveq1 6190	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
26		fveq1 6190	. . . . . 6 |- ( g = G -> ( g ` x ) = ( G ` x ) )
27	25, 26	oveqan12d 6669	. . . . 5 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) R ( g ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
28	24, 27	mpteq12dv 4733	. . . 4 |- ( ( f = F /\ g = G ) -> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
29		df-of 6897	. . . 4 |- oF R = ( f e. _V , g e. _V |-> ( x e. ( dom f i^i dom g ) |-> ( ( f ` x ) R ( g ` x ) ) ) )
30	28, 29	ovmpt2ga 6790	. . 3 |- ( ( F e. _V /\ G e. _V /\ ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V ) -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
31	4, 8, 21, 30	syl3anc 1326	. 2 |- ( ph -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
32	14	eleq2i 2693	. . . . 5 |- ( x e. ( A i^i B ) <-> x e. S )
33		elin 3796	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
34	32, 33	bitr3i 266	. . . 4 |- ( x e. S <-> ( x e. A /\ x e. B ) )
35		offval.6	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )
36	35	adantrr 753	. . . . 5 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( F ` x ) = C )
37		offval.7	. . . . . 6 |- ( ( ph /\ x e. B ) -> ( G ` x ) = D )
38	37	adantrl 752	. . . . 5 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( G ` x ) = D )
39	36, 38	oveq12d 6668	. . . 4 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( ( F ` x ) R ( G ` x ) ) = ( C R D ) )
40	34, 39	sylan2b 492	. . 3 |- ( ( ph /\ x e. S ) -> ( ( F ` x ) R ( G ` x ) ) = ( C R D ) )
41	40	mpteq2dva 4744	. 2 |- ( ph -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( C R D ) ) )
42	31, 16, 41	3eqtrd 2660	1 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   i^i cin 3573   |-> cmpt 4729   dom cdm 5114   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:  ofval  6906  offn  6908  offval2f  6909  off  6912  ofres  6913  offval2  6914  ofco  6917  offveqb  6919  suppssof1  7328  o1rlimmul  14349  gsumbagdiaglem  19375  evlslem1  19515  psrplusgpropd  19606  frlmipval  20118  frlmphllem  20119  frlmphl  20120  mat1dimscm  20281  rrxcph  23180  rrxds  23181  mbfadd  23428  mbfsub  23429  mbfmullem2  23491  mbfmul  23493  bddmulibl  23605  dvcmulf  23708  ofrn2  29442  off2  29443  ofresid  29444  ofcof  30169  plymul02  30623  signsplypnf  30627  signsply0  30628  matunitlindflem1  33405  matunitlindflem2  33406  poimirlem3  33412  poimirlem4  33413  poimirlem16  33425  poimirlem19  33428  poimirlem28  33437  broucube  33443  itg2addnc  33464  ftc1anclem8  33492  dflinc2  42199  fdivmpt  42334