Theorem ofrfval 6069

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 5718	. . . 4 |- ( ( F Fn A /\ A e. V ) -> F e. _V )
4	1, 2, 3	syl2anc 409	. . 3 |- ( ph -> F e. _V )
5		offval.2	. . . 4 |- ( ph -> G Fn B )
6		offval.4	. . . 4 |- ( ph -> B e. W )
7		fnex 5718	. . . 4 |- ( ( G Fn B /\ B e. W ) -> G e. _V )
8	5, 6, 7	syl2anc 409	. . 3 |- ( ph -> G e. _V )
9		dmeq 4811	. . . . . 6 |- ( f = F -> dom f = dom F )
10		dmeq 4811	. . . . . 6 |- ( g = G -> dom g = dom G )
11	9, 10	ineqan12d 3330	. . . . 5 |- ( ( f = F /\ g = G ) -> ( dom f i^i dom g ) = ( dom F i^i dom G ) )
12		fveq1 5495	. . . . . 6 |- ( f = F -> ( f ` x ) = ( F ` x ) )
13		fveq1 5495	. . . . . 6 |- ( g = G -> ( g ` x ) = ( G ` x ) )
14	12, 13	breqan12d 4005	. . . . 5 |- ( ( f = F /\ g = G ) -> ( ( f ` x ) R ( g ` x ) <-> ( F ` x ) R ( G ` x ) ) )
15	11, 14	raleqbidv 2677	. . . 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 6062	. . . 4 |- oR R = { <. f , g >. | A. x e. ( dom f i^i dom g ) ( f ` x ) R ( g ` x ) }
17	15, 16	brabga 4249	. . 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 409	. 2 |- ( ph -> ( F oR R G <-> A. x e. ( dom F i^i dom G ) ( F ` x ) R ( G ` x ) ) )
19		fndm 5297	. . . . . 6 |- ( F Fn A -> dom F = A )
20	1, 19	syl 14	. . . . 5 |- ( ph -> dom F = A )
21		fndm 5297	. . . . . 6 |- ( G Fn B -> dom G = B )
22	5, 21	syl 14	. . . . 5 |- ( ph -> dom G = B )
23	20, 22	ineq12d 3329	. . . 4 |- ( ph -> ( dom F i^i dom G ) = ( A i^i B ) )
24		offval.5	. . . 4 |- ( A i^i B ) = S
25	23, 24	eqtrdi 2219	. . 3 |- ( ph -> ( dom F i^i dom G ) = S )
26	25	raleqdv 2671	. 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 3347	. . . . . . 7 |- ( A i^i B ) C_ A
28	24, 27	eqsstrri 3180	. . . . . 6 |- S C_ A
29	28	sseli 3143	. . . . 5 |- ( x e. S -> x e. A )
30		offval.6	. . . . 5 |- ( ( ph /\ x e. A ) -> ( F ` x ) = C )
31	29, 30	sylan2 284	. . . 4 |- ( ( ph /\ x e. S ) -> ( F ` x ) = C )
32		inss2 3348	. . . . . . 7 |- ( A i^i B ) C_ B
33	24, 32	eqsstrri 3180	. . . . . 6 |- S C_ B
34	33	sseli 3143	. . . . 5 |- ( x e. S -> x e. B )
35		offval.7	. . . . 5 |- ( ( ph /\ x e. B ) -> ( G ` x ) = D )
36	34, 35	sylan2 284	. . . 4 |- ( ( ph /\ x e. S ) -> ( G ` x ) = D )
37	31, 36	breq12d 4002	. . 3 |- ( ( ph /\ x e. S ) -> ( ( F ` x ) R ( G ` x ) <-> C R D ) )
38	37	ralbidva 2466	. 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 1348   e. wcel 2141   A.wral 2448   _Vcvv 2730   i^i cin 3120   class class class wbr 3989   dom cdm 4611   Fn wfn 5193   ` cfv 5198   oRcofr 6060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ofr 6062

This theorem is referenced by:  ofrval  6071  ofrfval2  6077  caofref  6082  caofrss  6085  caoftrn  6086