Theorem ovi3 5900

Description: The value of an operation class abstraction. Special case. (Contributed by NM, 28-May-1995.) (Revised by Mario Carneiro, 29-Dec-2014.)

Hypotheses

Ref	Expression
ovi3.1	|- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> S e. ( H X. H ) )
ovi3.2	|- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> R = S )
ovi3.3	|- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }

Assertion

Ref	Expression
ovi3	|- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( <. A , B >. F <. C , D >. ) = S )

Distinct variable groups:   u, f, v, w, x, y, z, A   B, f, u, v, w, x, y, z   x, R, y, z   C, f, u, v, w, y, z   D, f, u, v, w, y, z   f, H, u, v, w, x, y, z   S, f, u, v, w, z

Allowed substitution hints:   C( x)   D( x)   R( w, v, u, f)   S( x, y)   F( x, y, z, w, v, u, f)

Proof of Theorem ovi3

Step	Hyp	Ref	Expression
1		ovi3.1	. . . 4 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> S e. ( H X. H ) )
2		elex 2692	. . . 4 |- ( S e. ( H X. H ) -> S e. _V )
3	1, 2	syl 14	. . 3 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> S e. _V )
4		isset 2687	. . 3 |- ( S e. _V <-> E. z z = S )
5	3, 4	sylib 121	. 2 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> E. z z = S )
6		nfv 1508	. . 3 |- F/ z ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) )
7		nfcv 2279	. . . . 5 |- F/_ z <. A , B >.
8		ovi3.3	. . . . . 6 |- F = { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
9		nfoprab3 5815	. . . . . 6 |- F/_ z { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
10	8, 9	nfcxfr 2276	. . . . 5 |- F/_ z F
11		nfcv 2279	. . . . 5 |- F/_ z <. C , D >.
12	7, 10, 11	nfov 5794	. . . 4 |- F/_ z ( <. A , B >. F <. C , D >. )
13	12	nfeq1 2289	. . 3 |- F/ z ( <. A , B >. F <. C , D >. ) = S
14		ovi3.2	. . . . . . 7 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> R = S )
15	14	eqeq2d 2149	. . . . . 6 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( z = R <-> z = S ) )
16	15	copsex4g 4164	. . . . 5 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) <-> z = S ) )
17		opelxpi 4566	. . . . . 6 |- ( ( A e. H /\ B e. H ) -> <. A , B >. e. ( H X. H ) )
18		opelxpi 4566	. . . . . 6 |- ( ( C e. H /\ D e. H ) -> <. C , D >. e. ( H X. H ) )
19		nfcv 2279	. . . . . . 7 |- F/_ x <. A , B >.
20		nfcv 2279	. . . . . . 7 |- F/_ y <. A , B >.
21		nfcv 2279	. . . . . . 7 |- F/_ y <. C , D >.
22		nfv 1508	. . . . . . . 8 |- F/ x E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R )
23		nfoprab1 5813	. . . . . . . . . . 11 |- F/_ x { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
24	8, 23	nfcxfr 2276	. . . . . . . . . 10 |- F/_ x F
25		nfcv 2279	. . . . . . . . . 10 |- F/_ x y
26	19, 24, 25	nfov 5794	. . . . . . . . 9 |- F/_ x ( <. A , B >. F y )
27	26	nfeq1 2289	. . . . . . . 8 |- F/ x ( <. A , B >. F y ) = z
28	22, 27	nfim 1551	. . . . . . 7 |- F/ x ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F y ) = z )
29		nfv 1508	. . . . . . . 8 |- F/ y E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R )
30		nfoprab2 5814	. . . . . . . . . . 11 |- F/_ y { <. <. x , y >. , z >. | ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) /\ E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) }
31	8, 30	nfcxfr 2276	. . . . . . . . . 10 |- F/_ y F
32	20, 31, 21	nfov 5794	. . . . . . . . 9 |- F/_ y ( <. A , B >. F <. C , D >. )
33	32	nfeq1 2289	. . . . . . . 8 |- F/ y ( <. A , B >. F <. C , D >. ) = z
34	29, 33	nfim 1551	. . . . . . 7 |- F/ y ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F <. C , D >. ) = z )
35		eqeq1 2144	. . . . . . . . . . 11 |- ( x = <. A , B >. -> ( x = <. w , v >. <-> <. A , B >. = <. w , v >. ) )
36	35	anbi1d 460	. . . . . . . . . 10 |- ( x = <. A , B >. -> ( ( x = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) ) )
37	36	anbi1d 460	. . . . . . . . 9 |- ( x = <. A , B >. -> ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) )
38	37	4exbidv 1842	. . . . . . . 8 |- ( x = <. A , B >. -> ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) )
39		oveq1 5774	. . . . . . . . 9 |- ( x = <. A , B >. -> ( x F y ) = ( <. A , B >. F y ) )
40	39	eqeq1d 2146	. . . . . . . 8 |- ( x = <. A , B >. -> ( ( x F y ) = z <-> ( <. A , B >. F y ) = z ) )
41	38, 40	imbi12d 233	. . . . . . 7 |- ( x = <. A , B >. -> ( ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( x F y ) = z ) <-> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F y ) = z ) ) )
42		eqeq1 2144	. . . . . . . . . . 11 |- ( y = <. C , D >. -> ( y = <. u , f >. <-> <. C , D >. = <. u , f >. ) )
43	42	anbi2d 459	. . . . . . . . . 10 |- ( y = <. C , D >. -> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) ) )
44	43	anbi1d 460	. . . . . . . . 9 |- ( y = <. C , D >. -> ( ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) ) )
45	44	4exbidv 1842	. . . . . . . 8 |- ( y = <. C , D >. -> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) ) )
46		oveq2 5775	. . . . . . . . 9 |- ( y = <. C , D >. -> ( <. A , B >. F y ) = ( <. A , B >. F <. C , D >. ) )
47	46	eqeq1d 2146	. . . . . . . 8 |- ( y = <. C , D >. -> ( ( <. A , B >. F y ) = z <-> ( <. A , B >. F <. C , D >. ) = z ) )
48	45, 47	imbi12d 233	. . . . . . 7 |- ( y = <. C , D >. -> ( ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F y ) = z ) <-> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F <. C , D >. ) = z ) ) )
49		moeq 2854	. . . . . . . . . . . 12 |- E* z z = R
50	49	mosubop 4600	. . . . . . . . . . 11 |- E* z E. u E. f ( y = <. u , f >. /\ z = R )
51	50	mosubop 4600	. . . . . . . . . 10 |- E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) )
52		anass 398	. . . . . . . . . . . . . 14 |- ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) )
53	52	2exbii 1585	. . . . . . . . . . . . 13 |- ( E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E. u E. f ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) )
54		19.42vv 1883	. . . . . . . . . . . . 13 |- ( E. u E. f ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) <-> ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) ) )
55	53, 54	bitri 183	. . . . . . . . . . . 12 |- ( E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) ) )
56	55	2exbii 1585	. . . . . . . . . . 11 |- ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) ) )
57	56	mobii 2034	. . . . . . . . . 10 |- ( E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) ) )
58	51, 57	mpbir 145	. . . . . . . . 9 |- E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R )
59	58	a1i 9	. . . . . . . 8 |- ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) -> E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) )
60	59, 8	ovidi 5882	. . . . . . 7 |- ( ( x e. ( H X. H ) /\ y e. ( H X. H ) ) -> ( E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) -> ( x F y ) = z ) )
61	19, 20, 21, 28, 34, 41, 48, 60	vtocl2gaf 2748	. . . . . 6 |- ( ( <. A , B >. e. ( H X. H ) /\ <. C , D >. e. ( H X. H ) ) -> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F <. C , D >. ) = z ) )
62	17, 18, 61	syl2an 287	. . . . 5 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) -> ( <. A , B >. F <. C , D >. ) = z ) )
63	16, 62	sylbird 169	. . . 4 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( z = S -> ( <. A , B >. F <. C , D >. ) = z ) )
64		eqeq2 2147	. . . 4 |- ( z = S -> ( ( <. A , B >. F <. C , D >. ) = z <-> ( <. A , B >. F <. C , D >. ) = S ) )
65	63, 64	mpbidi 150	. . 3 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( z = S -> ( <. A , B >. F <. C , D >. ) = S ) )
66	6, 13, 65	exlimd 1576	. 2 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( E. z z = S -> ( <. A , B >. F <. C , D >. ) = S ) )
67	5, 66	mpd 13	1 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( <. A , B >. F <. C , D >. ) = S )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   E.wex 1468   e. wcel 1480   E*wmo 1998   _Vcvv 2681   <.cop 3525   X. cxp 4532  (class class class)co 5767   {coprab 5768

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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-setind 4447

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771

This theorem is referenced by:  oviec  6528  addcnsr  7635  mulcnsr  7636