Theorem ofrfval 5922

Description: Value of a relation applied to two functions. (Contributed by Mario Carneiro, 28-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
ofrfval	|- ( ph -> ( F oR R G <-> A. 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 ofrfval

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 5574	. . . 4 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
4	1, 2, 3	syl2anc 406	. . 3 |- ( ph -> F e. _V )
5		offval.2	. . . 4 |- ( ph -> G Fn B )
6		offval.4	. . . 4 |- ( ph -> B e. W )
7		fnex 5574	. . . 4 |- ( ( G Fn B /\ B e. W ) -> G e. _V )
8	5, 6, 7	syl2anc 406	. . 3 |- ( ph -> G e. _V )
9		dmeq 4677	. . . . . 6 |- ( f = F -> dom f = dom F )
10		dmeq 4677	. . . . . 6 |- ( g = G -> dom g = dom G )
11	9, 10	ineqan12d 3226	. . . . 5 |- ( ( f = F /\ g = G ) -> ( dom f i^i dom g ) = ( dom F i^i dom G ) )
12		fveq1 5352	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
13		fveq1 5352	. . . . . 6 |- ( g = G -> ( g ` x ) = ( G ` x ) )
14	12, 13	breqan12d 3890	. . . . 5 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) R ( g ` x ) <-> ( F ` x ) R ( G ` x ) ) )
15	11, 14	raleqbidv 2596	. . . 4 |- ( ( f = F /\ g = G ) -> ( A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) <-> A. x e. ( dom F i^i dom G ) ( F ` x ) R ( G ` x ) ) )
16		df-ofr 5915	. . . 4 |- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
17	15, 16	brabga 4124	. . 3 |- ( ( F e. _V /\ G e. _V ) -> ( F oR R G <-> A. x e. ( dom F i^i dom G ) ( F ` x ) R ( G ` x ) ) )
18	4, 8, 17	syl2anc 406	. 2 |- ( ph -> ( F oR R G <-> A. x e. ( dom F i^i dom G ) ( F ` x ) R ( G ` x ) ) )
19		fndm 5158	. . . . . 6 |- ( F Fn A -> dom F = A )
20	1, 19	syl 14	. . . . 5 |- ( ph -> dom F = A )
21		fndm 5158	. . . . . 6 |- ( G Fn B -> dom G = B )
22	5, 21	syl 14	. . . . 5 |- ( ph -> dom G = B )
23	20, 22	ineq12d 3225	. . . 4 |- ( ph -> ( dom F i^i dom G ) = ( A i^i B ) )
24		offval.5	. . . 4 |- ( A i^i B ) = S
25	23, 24	syl6eq 2148	. . 3 |- ( ph -> ( dom F i^i dom G ) = S )
26	25	raleqdv 2590	. 2 |- ( ph -> ( A. x e. ( dom F i^i dom G ) ( F ` x ) R ( G ` x ) <-> A. x e. S ( F ` x ) R ( G ` x ) ) )
27		inss1 3243	. . . . . . 7 |- ( A i^i B ) C_ A
28	24, 27	eqsstr3i 3080	. . . . . 6 |- S C_ A
29	28	sseli 3043	. . . . 5 |- ( x e. S -> x e. A )
30		offval.6	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )
31	29, 30	sylan2 282	. . . 4 |- ( ( ph /\ x e. S ) -> ( F ` x ) = C )
32		inss2 3244	. . . . . . 7 |- ( A i^i B ) C_ B
33	24, 32	eqsstr3i 3080	. . . . . 6 |- S C_ B
34	33	sseli 3043	. . . . 5 |- ( x e. S -> x e. B )
35		offval.7	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = D )
36	34, 35	sylan2 282	. . . 4 |- ( ( ph /\ x e. S ) -> ( G ` x ) = D )
37	31, 36	breq12d 3888	. . 3 |- ( ( ph /\ x e. S ) -> ( ( F ` x ) R ( G ` x ) <-> C R D ) )
38	37	ralbidva 2392	. 2 |- ( ph -> ( A. x e. S ( F ` x ) R ( G ` x ) <-> A. x e. S C R D ) )
39	18, 26, 38	3bitrd 213	1 |- ( ph -> ( F oR R G <-> A. x e. S C R D ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   A.wral 2375   _Vcvv 2641   i^i cin 3020   class class class wbr 3875   dom cdm 4477   Fn wfn 5054   ` cfv 5059   oRcofr 5913

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-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-coll 3983  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-iun 3762  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-f1 5064  df-fo 5065  df-f1o 5066  df-fv 5067  df-ofr 5915

This theorem is referenced by:  ofrval  5924  ofrfval2  5929  caofref  5934  caofrss  5937  caoftrn  5938