Theorem offval 5901

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 5558	. . . 4 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
4	1, 2, 3	syl2anc 404	. . 3 |- ( ph -> F e. _V )
5		offval.2	. . . 4 |- ( ph -> G Fn B )
6		offval.4	. . . 4 |- ( ph -> B e. W )
7		fnex 5558	. . . 4 |- ( ( G Fn B /\ B e. W ) -> G e. _V )
8	5, 6, 7	syl2anc 404	. . 3 |- ( ph -> G e. _V )
9		fndm 5147	. . . . . . . 8 |- ( F Fn A -> dom F = A )
10	1, 9	syl 14	. . . . . . 7 |- ( ph -> dom F = A )
11		fndm 5147	. . . . . . . 8 |- ( G Fn B -> dom G = B )
12	5, 11	syl 14	. . . . . . 7 |- ( ph -> dom G = B )
13	10, 12	ineq12d 3217	. . . . . 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 2143	. . . . 5 |- ( ph -> ( dom F i^i dom G ) = S )
16	15	mpteq1d 3945	. . . 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 3996	. . . . . 6 |- ( A e. V -> ( A i^i B ) e. _V )
18	14, 17	syl5eqelr 2182	. . . . 5 |- ( A e. V -> S e. _V )
19		mptexg 5561	. . . . 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 2171	. . 3 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
22		dmeq 4667	. . . . . 6 |- ( f = F -> dom f = dom F )
23		dmeq 4667	. . . . . 6 |- ( g = G -> dom g = dom G )
24	22, 23	ineqan12d 3218	. . . . 5 |- ( ( f = F /\ g = G ) -> ( dom f i^i dom g ) = ( dom F i^i dom G ) )
25		fveq1 5339	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
26		fveq1 5339	. . . . . 6 |- ( g = G -> ( g ` x ) = ( G ` x ) )
27	25, 26	oveqan12d 5709	. . . . 5 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) R ( g ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
28	24, 27	mpteq12dv 3942	. . . 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 5894	. . . 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 5812	. . 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 1181	. 2 |- ( ph -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
32	14	eleq2i 2161	. . . . 5 |- ( x e. ( A i^i B ) <-> x e. S )
33		elin 3198	. . . . 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 464	. . . . 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 463	. . . . 5 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( G ` x ) = D )
39	36, 38	oveq12d 5708	. . . 4 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( ( F ` x ) R ( G ` x ) ) = ( C R D ) )
40	34, 39	sylan2b 282	. . 3 |- ( ( ph /\ x e. S ) -> ( ( F ` x ) R ( G ` x ) ) = ( C R D ) )
41	40	mpteq2dva 3950	. 2 |- ( ph -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( C R D ) ) )
42	31, 16, 41	3eqtrd 2131	1 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1296   e. wcel 1445   _Vcvv 2633   i^i cin 3012   |-> cmpt 3921   dom cdm 4467   Fn wfn 5044   ` cfv 5049  (class class class)co 5690   oFcof 5892

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-coll 3975  ax-sep 3978  ax-pow 4030  ax-pr 4060  ax-setind 4381

This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-ral 2375  df-rex 2376  df-reu 2377  df-rab 2379  df-v 2635  df-sbc 2855  df-csb 2948  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-iun 3754  df-br 3868  df-opab 3922  df-mpt 3923  df-id 4144  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-res 4479  df-ima 4480  df-iota 5014  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-fv 5057  df-ov 5693  df-oprab 5694  df-mpt2 5695  df-of 5894

This theorem is referenced by:  fnofval  5903  off  5906  ofres  5907  offval2  5908  suppssof1  5910  ofco  5911  offveqb  5912