Theorem ofrfval 5740

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 5404	. . . 4 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
4	1, 2, 3	syl2anc 403	. . 3 |- ( ph -> F e. _V )
5		offval.2	. . . 4 |- ( ph -> G Fn B )
6		offval.4	. . . 4 |- ( ph -> B e. W )
7		fnex 5404	. . . 4 |- ( ( G Fn B /\ B e. W ) -> G e. _V )
8	5, 6, 7	syl2anc 403	. . 3 |- ( ph -> G e. _V )
9		dmeq 4553	. . . . . 6 |- ( f = F -> dom f = dom F )
10		dmeq 4553	. . . . . 6 |- ( g = G -> dom g = dom G )
11	9, 10	ineqan12d 3169	. . . . 5 |- ( ( f = F /\ g = G ) -> ( dom f i^i dom g ) = ( dom F i^i dom G ) )
12		fveq1 5197	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
13		fveq1 5197	. . . . . 6 |- ( g = G -> ( g ` x ) = ( G ` x ) )
14	12, 13	breqan12d 3800	. . . . 5 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) R ( g ` x ) <-> ( F ` x ) R ( G ` x ) ) )
15	11, 14	raleqbidv 2561	. . . 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 5733	. . . 4 |- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
17	15, 16	brabga 4019	. . 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 403	. 2 |- ( ph -> ( F oR R G <-> A. x e. ( dom F i^i dom G ) ( F ` x ) R ( G ` x ) ) )
19		fndm 5018	. . . . . 6 |- ( F Fn A -> dom F = A )
20	1, 19	syl 14	. . . . 5 |- ( ph -> dom F = A )
21		fndm 5018	. . . . . 6 |- ( G Fn B -> dom G = B )
22	5, 21	syl 14	. . . . 5 |- ( ph -> dom G = B )
23	20, 22	ineq12d 3168	. . . 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 2129	. . 3 |- ( ph -> ( dom F i^i dom G ) = S )
26	25	raleqdv 2555	. 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 3186	. . . . . . 7 |- ( A i^i B ) C_ A
28	24, 27	eqsstr3i 3030	. . . . . 6 |- S C_ A
29	28	sseli 2995	. . . . 5 |- ( x e. S -> x e. A )
30		offval.6	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )
31	29, 30	sylan2 280	. . . 4 |- ( ( ph /\ x e. S ) -> ( F ` x ) = C )
32		inss2 3187	. . . . . . 7 |- ( A i^i B ) C_ B
33	24, 32	eqsstr3i 3030	. . . . . 6 |- S C_ B
34	33	sseli 2995	. . . . 5 |- ( x e. S -> x e. B )
35		offval.7	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = D )
36	34, 35	sylan2 280	. . . 4 |- ( ( ph /\ x e. S ) -> ( G ` x ) = D )
37	31, 36	breq12d 3798	. . 3 |- ( ( ph /\ x e. S ) -> ( ( F ` x ) R ( G ` x ) <-> C R D ) )
38	37	ralbidva 2364	. 2 |- ( ph -> ( A. x e. S ( F ` x ) R ( G ` x ) <-> A. x e. S C R D ) )
39	18, 26, 38	3bitrd 212	1 |- ( ph -> ( F oR R G <-> A. x e. S C R D ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   _Vcvv 2601   i^i cin 2972   class class class wbr 3785   dom cdm 4363   Fn wfn 4917   ` cfv 4922   oRcofr 5731

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-ofr 5733

This theorem is referenced by:  ofrval  5742  ofrfval2  5747  caofref  5752  caofrss  5755  caoftrn  5756