Theorem offval 6138

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 5780	. . . 4 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
4	1, 2, 3	syl2anc 411	. . 3 |- ( ph -> F e. _V )
5		offval.2	. . . 4 |- ( ph -> G Fn B )
6		offval.4	. . . 4 |- ( ph -> B e. W )
7		fnex 5780	. . . 4 |- ( ( G Fn B /\ B e. W ) -> G e. _V )
8	5, 6, 7	syl2anc 411	. . 3 |- ( ph -> G e. _V )
9		fndm 5353	. . . . . . . 8 |- ( F Fn A -> dom F = A )
10	1, 9	syl 14	. . . . . . 7 |- ( ph -> dom F = A )
11		fndm 5353	. . . . . . . 8 |- ( G Fn B -> dom G = B )
12	5, 11	syl 14	. . . . . . 7 |- ( ph -> dom G = B )
13	10, 12	ineq12d 3361	. . . . . 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 2242	. . . . 5 |- ( ph -> ( dom F i^i dom G ) = S )
16	15	mpteq1d 4114	. . . 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 4165	. . . . . 6 |- ( A e. V -> ( A i^i B ) e. _V )
18	14, 17	eqeltrrid 2281	. . . . 5 |- ( A e. V -> S e. _V )
19		mptexg 5783	. . . . 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 2270	. . 3 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
22		dmeq 4862	. . . . . 6 |- ( f = F -> dom f = dom F )
23		dmeq 4862	. . . . . 6 |- ( g = G -> dom g = dom G )
24	22, 23	ineqan12d 3362	. . . . 5 |- ( ( f = F /\ g = G ) -> ( dom f i^i dom g ) = ( dom F i^i dom G ) )
25		fveq1 5553	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
26		fveq1 5553	. . . . . 6 |- ( g = G -> ( g ` x ) = ( G ` x ) )
27	25, 26	oveqan12d 5937	. . . . 5 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) R ( g ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
28	24, 27	mpteq12dv 4111	. . . 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 6130	. . . 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 6048	. . 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 1249	. 2 |- ( ph -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
32	14	eleq2i 2260	. . . . 5 |- ( x e. ( A i^i B ) <-> x e. S )
33		elin 3342	. . . . 5 |- ( x e. ( A i^i B ) <-> ( x e. A /\ x e. B ) )
34	32, 33	bitr3i 186	. . . 4 |- ( x e. S <-> ( x e. A /\ x e. B ) )
35		offval.6	. . . . . 6 |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )
36	35	adantrr 479	. . . . 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 478	. . . . 5 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( G ` x ) = D )
39	36, 38	oveq12d 5936	. . . 4 |- ( ( ph /\ ( x e. A /\ x e. B ) ) -> ( ( F ` x ) R ( G ` x ) ) = ( C R D ) )
40	34, 39	sylan2b 287	. . 3 |- ( ( ph /\ x e. S ) -> ( ( F ` x ) R ( G ` x ) ) = ( C R D ) )
41	40	mpteq2dva 4119	. 2 |- ( ph -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( C R D ) ) )
42	31, 16, 41	3eqtrd 2230	1 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   i^i cin 3152   |-> cmpt 4090   dom cdm 4659   Fn wfn 5249   ` cfv 5254  (class class class)co 5918   oFcof 6128

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-setind 4569

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-ov 5921  df-oprab 5922  df-mpo 5923  df-of 6130

This theorem is referenced by:  ofvalg  6140  off  6143  ofres  6145  offval2  6146  suppssof1  6148  ofco  6149  offveqb  6150