Theorem offval 6057

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 5707	. . . 4 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
4	1, 2, 3	syl2anc 409	. . 3 |- ( ph -> F e. _V )
5		offval.2	. . . 4 |- ( ph -> G Fn B )
6		offval.4	. . . 4 |- ( ph -> B e. W )
7		fnex 5707	. . . 4 |- ( ( G Fn B /\ B e. W ) -> G e. _V )
8	5, 6, 7	syl2anc 409	. . 3 |- ( ph -> G e. _V )
9		fndm 5287	. . . . . . . 8 |- ( F Fn A -> dom F = A )
10	1, 9	syl 14	. . . . . . 7 |- ( ph -> dom F = A )
11		fndm 5287	. . . . . . . 8 |- ( G Fn B -> dom G = B )
12	5, 11	syl 14	. . . . . . 7 |- ( ph -> dom G = B )
13	10, 12	ineq12d 3324	. . . . . 6 |- ( ph -> ( dom F i^i dom G ) = ( A i^i B ) )
14		offval.5	. . . . . 6 |- ( A i^i B ) = S
15	13, 14	eqtrdi 2215	. . . . 5 |- ( ph -> ( dom F i^i dom G ) = S )
16	15	mpteq1d 4067	. . . 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 4118	. . . . . 6 |- ( A e. V -> ( A i^i B ) e. _V )
18	14, 17	eqeltrrid 2254	. . . . 5 |- ( A e. V -> S e. _V )
19		mptexg 5710	. . . . 5 |- ( S e. _V -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
20	2, 18, 19	3syl 17	. . . 4 |- ( ph -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
21	16, 20	eqeltrd 2243	. . 3 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
22		dmeq 4804	. . . . . 6 |- ( f = F -> dom f = dom F )
23		dmeq 4804	. . . . . 6 |- ( g = G -> dom g = dom G )
24	22, 23	ineqan12d 3325	. . . . 5 |- ( ( f = F /\ g = G ) -> ( dom f i^i dom g ) = ( dom F i^i dom G ) )
25		fveq1 5485	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
26		fveq1 5485	. . . . . 6 |- ( g = G -> ( g ` x ) = ( G ` x ) )
27	25, 26	oveqan12d 5861	. . . . 5 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) R ( g ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
28	24, 27	mpteq12dv 4064	. . . 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 6050	. . . 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	ovmpoga 5971	. . 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 1228	. 2 |- ( ph -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
32	14	eleq2i 2233	. . . . 5 |- ( x e. ( A i^i B ) <-> x e. S )
33		elin 3305	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
34	32, 33	bitr3i 185	. . . 4 |- ( x e. S <-> ( x e. A /\ x e. B ) )
35		offval.6	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )
36	35	adantrr 471	. . . . 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 470	. . . . 5 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( G ` x ) = D )
39	36, 38	oveq12d 5860	. . . 4 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( ( F ` x ) R ( G ` x ) ) = ( C R D ) )
40	34, 39	sylan2b 285	. . 3 |- ( ( ph /\ x e. S ) -> ( ( F ` x ) R ( G ` x ) ) = ( C R D ) )
41	40	mpteq2dva 4072	. 2 |- ( ph -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( C R D ) ) )
42	31, 16, 41	3eqtrd 2202	1 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   _Vcvv 2726   i^i cin 3115   |-> cmpt 4043   dom cdm 4604   Fn wfn 5183   ` cfv 5188  (class class class)co 5842   oFcof 6048

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050

This theorem is referenced by:  ofvalg  6059  off  6062  ofres  6064  offval2  6065  suppssof1  6067  ofco  6068  offveqb  6069