Theorem offval 6189

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 5829	. . . 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 5829	. . . 4 |- ( ( G Fn B /\ B e. W ) -> G e. _V )
8	5, 6, 7	syl2anc 411	. . 3 |- ( ph -> G e. _V )
9		fndm 5392	. . . . . . . 8 |- ( F Fn A -> dom F = A )
10	1, 9	syl 14	. . . . . . 7 |- ( ph -> dom F = A )
11		fndm 5392	. . . . . . . 8 |- ( G Fn B -> dom G = B )
12	5, 11	syl 14	. . . . . . 7 |- ( ph -> dom G = B )
13	10, 12	ineq12d 3383	. . . . . 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 2256	. . . . 5 |- ( ph -> ( dom F i^i dom G ) = S )
16	15	mpteq1d 4145	. . . 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 4196	. . . . . 6 |- ( A e. V -> ( A i^i B ) e. _V )
18	14, 17	eqeltrrid 2295	. . . . 5 |- ( A e. V -> S e. _V )
19		mptexg 5832	. . . . 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 2284	. . 3 |- ( ph -> ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) e. _V )
22		dmeq 4897	. . . . . 6 |- ( f = F -> dom f = dom F )
23		dmeq 4897	. . . . . 6 |- ( g = G -> dom g = dom G )
24	22, 23	ineqan12d 3384	. . . . 5 |- ( ( f = F /\ g = G ) -> ( dom f i^i dom g ) = ( dom F i^i dom G ) )
25		fveq1 5598	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
26		fveq1 5598	. . . . . 6 |- ( g = G -> ( g ` x ) = ( G ` x ) )
27	25, 26	oveqan12d 5986	. . . . 5 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) R ( g ` x ) ) = ( ( F ` x ) R ( G ` x ) ) )
28	24, 27	mpteq12dv 4142	. . . 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 6181	. . . 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 6098	. . 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 1250	. 2 |- ( ph -> ( F oF R G ) = ( x e. ( dom F i^i dom G ) |-> ( ( F ` x ) R ( G ` x ) ) ) )
32	14	eleq2i 2274	. . . . 5 |- ( x e. ( A i^i B ) <-> x e. S )
33		elin 3364	. . . . 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 5985	. . . 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 4150	. 2 |- ( ph -> ( x e. S |-> ( ( F ` x ) R ( G ` x ) ) ) = ( x e. S |-> ( C R D ) ) )
42	31, 16, 41	3eqtrd 2244	1 |- ( ph -> ( F oF R G ) = ( x e. S |-> ( C R D ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1373   e. wcel 2178   _Vcvv 2776   i^i cin 3173   |-> cmpt 4121   dom cdm 4693   Fn wfn 5285   ` cfv 5290  (class class class)co 5967   oFcof 6179

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-of 6181

This theorem is referenced by:  ofvalg  6191  off  6194  ofres  6196  offval2  6197  suppssof1  6199  ofco  6200  offveqb  6201