Theorem ovi3 5989

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 2741	. . . 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 2736	. . 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 1521	. . 3 |- F/ z ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) )
7		nfcv 2312	. . . . 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 5904	. . . . . 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 2309	. . . . 5 |- F/_ z F
11		nfcv 2312	. . . . 5 |- F/_ z <. C , D >.
12	7, 10, 11	nfov 5883	. . . 4 |- F/_ z ( <. A , B >. F <. C , D >. )
13	12	nfeq1 2322	. . 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 2182	. . . . . 6 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( z = R <-> z = S ) )
16	15	copsex4g 4232	. . . . 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 4643	. . . . . 6 |- ( ( A e. H /\ B e. H ) -> <. A , B >. e. ( H X. H ) )
18		opelxpi 4643	. . . . . 6 |- ( ( C e. H /\ D e. H ) -> <. C , D >. e. ( H X. H ) )
19		nfcv 2312	. . . . . . 7 |- F/_ x <. A , B >.
20		nfcv 2312	. . . . . . 7 |- F/_ y <. A , B >.
21		nfcv 2312	. . . . . . 7 |- F/_ y <. C , D >.
22		nfv 1521	. . . . . . . 8 |- F/ x E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R )
23		nfoprab1 5902	. . . . . . . . . . 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 2309	. . . . . . . . . 10 |- F/_ x F
25		nfcv 2312	. . . . . . . . . 10 |- F/_ x y
26	19, 24, 25	nfov 5883	. . . . . . . . 9 |- F/_ x ( <. A , B >. F y )
27	26	nfeq1 2322	. . . . . . . 8 |- F/ x ( <. A , B >. F y ) = z
28	22, 27	nfim 1565	. . . . . . 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 1521	. . . . . . . 8 |- F/ y E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R )
30		nfoprab2 5903	. . . . . . . . . . 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 2309	. . . . . . . . . 10 |- F/_ y F
32	20, 31, 21	nfov 5883	. . . . . . . . 9 |- F/_ y ( <. A , B >. F <. C , D >. )
33	32	nfeq1 2322	. . . . . . . 8 |- F/ y ( <. A , B >. F <. C , D >. ) = z
34	29, 33	nfim 1565	. . . . . . 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 2177	. . . . . . . . . . 11 |- ( x = <. A , B >. -> ( x = <. w , v >. <-> <. A , B >. = <. w , v >. ) )
36	35	anbi1d 462	. . . . . . . . . 10 |- ( x = <. A , B >. -> ( ( x = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) ) )
37	36	anbi1d 462	. . . . . . . . 9 |- ( x = <. A , B >. -> ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) )
38	37	4exbidv 1863	. . . . . . . 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 5860	. . . . . . . . 9 |- ( x = <. A , B >. -> ( x F y ) = ( <. A , B >. F y ) )
40	39	eqeq1d 2179	. . . . . . . 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 2177	. . . . . . . . . . 11 |- ( y = <. C , D >. -> ( y = <. u , f >. <-> <. C , D >. = <. u , f >. ) )
43	42	anbi2d 461	. . . . . . . . . 10 |- ( y = <. C , D >. -> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) ) )
44	43	anbi1d 462	. . . . . . . . 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 1863	. . . . . . . 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 5861	. . . . . . . . 9 |- ( y = <. C , D >. -> ( <. A , B >. F y ) = ( <. A , B >. F <. C , D >. ) )
47	46	eqeq1d 2179	. . . . . . . 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 2905	. . . . . . . . . . . 12 |- E* z z = R
50	49	mosubop 4677	. . . . . . . . . . 11 |- E* z E. u E. f ( y = <. u , f >. /\ z = R )
51	50	mosubop 4677	. . . . . . . . . 10 |- E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) )
52		anass 399	. . . . . . . . . . . . . 14 |- ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) )
53	52	2exbii 1599	. . . . . . . . . . . . 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 1904	. . . . . . . . . . . . 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 1599	. . . . . . . . . . 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 2056	. . . . . . . . . 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 5971	. . . . . . 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 2797	. . . . . 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 2180	. . . 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 1590	. 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 1348   E.wex 1485   E*wmo 2020   e. wcel 2141   _Vcvv 2730   <.cop 3586   X. cxp 4609  (class class class)co 5853   {coprab 5854

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 609  ax-in2 610  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-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  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-ne 2341  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  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-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857

This theorem is referenced by:  oviec  6619  addcnsr  7796  mulcnsr  7797