Theorem ov3 6421

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
ov3.1	|- S e. _V
ov3.2	|- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> R = S )
ov3.3	|- 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
ov3	|- ( ( ( 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 ov3

Step	Hyp	Ref	Expression
1		ov3.1	. . 3 |- S e. _V
2	1	isseti 3119	. 2 |- E. z z = S
3		nfv 1683	. . 3 |- F/ z ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) )
4		nfcv 2629	. . . . 5 |- F/_ z <. A , B >.
5		ov3.3	. . . . . 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 ) ) }
6		nfoprab3 6330	. . . . . 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 ) ) }
7	5, 6	nfcxfr 2627	. . . . 5 |- F/_ z F
8		nfcv 2629	. . . . 5 |- F/_ z <. C , D >.
9	4, 7, 8	nfov 6305	. . . 4 |- F/_ z ( <. A , B >. F <. C , D >. )
10	9	nfeq1 2644	. . 3 |- F/ z ( <. A , B >. F <. C , D >. ) = S
11		ov3.2	. . . . . . 7 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> R = S )
12	11	eqeq2d 2481	. . . . . 6 |- ( ( ( w = A /\ v = B ) /\ ( u = C /\ f = D ) ) -> ( z = R <-> z = S ) )
13	12	copsex4g 4736	. . . . 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 ) )
14		opelxpi 5030	. . . . . 6 |- ( ( A e. H /\ B e. H ) -> <. A , B >. e. ( H X. H ) )
15		opelxpi 5030	. . . . . 6 |- ( ( C e. H /\ D e. H ) -> <. C , D >. e. ( H X. H ) )
16		nfcv 2629	. . . . . . 7 |- F/_ x <. A , B >.
17		nfcv 2629	. . . . . . 7 |- F/_ y <. A , B >.
18		nfcv 2629	. . . . . . 7 |- F/_ y <. C , D >.
19		nfv 1683	. . . . . . . 8 |- F/ x E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R )
20		nfoprab1 6328	. . . . . . . . . . 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 ) ) }
21	5, 20	nfcxfr 2627	. . . . . . . . . 10 |- F/_ x F
22		nfcv 2629	. . . . . . . . . 10 |- F/_ x y
23	16, 21, 22	nfov 6305	. . . . . . . . 9 |- F/_ x ( <. A , B >. F y )
24	23	nfeq1 2644	. . . . . . . 8 |- F/ x ( <. A , B >. F y ) = z
25	19, 24	nfim 1867	. . . . . . 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 )
26		nfv 1683	. . . . . . . 8 |- F/ y E. w E. v E. u E. f ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R )
27		nfoprab2 6329	. . . . . . . . . . 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 ) ) }
28	5, 27	nfcxfr 2627	. . . . . . . . . 10 |- F/_ y F
29	17, 28, 18	nfov 6305	. . . . . . . . 9 |- F/_ y ( <. A , B >. F <. C , D >. )
30	29	nfeq1 2644	. . . . . . . 8 |- F/ y ( <. A , B >. F <. C , D >. ) = z
31	26, 30	nfim 1867	. . . . . . 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 )
32		eqeq1 2471	. . . . . . . . . . 11 |- ( x = <. A , B >. -> ( x = <. w , v >. <-> <. A , B >. = <. w , v >. ) )
33	32	anbi1d 704	. . . . . . . . . 10 |- ( x = <. A , B >. -> ( ( x = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) ) )
34	33	anbi1d 704	. . . . . . . . 9 |- ( x = <. A , B >. -> ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) ) )
35	34	4exbidv 1694	. . . . . . . 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 ) ) )
36		oveq1 6289	. . . . . . . . 9 |- ( x = <. A , B >. -> ( x F y ) = ( <. A , B >. F y ) )
37	36	eqeq1d 2469	. . . . . . . 8 |- ( x = <. A , B >. -> ( ( x F y ) = z <-> ( <. A , B >. F y ) = z ) )
38	35, 37	imbi12d 320	. . . . . . 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 ) ) )
39		eqeq1 2471	. . . . . . . . . . 11 |- ( y = <. C , D >. -> ( y = <. u , f >. <-> <. C , D >. = <. u , f >. ) )
40	39	anbi2d 703	. . . . . . . . . 10 |- ( y = <. C , D >. -> ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) <-> ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) ) )
41	40	anbi1d 704	. . . . . . . . 9 |- ( y = <. C , D >. -> ( ( ( <. A , B >. = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( ( <. A , B >. = <. w , v >. /\ <. C , D >. = <. u , f >. ) /\ z = R ) ) )
42	41	4exbidv 1694	. . . . . . . 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 ) ) )
43		oveq2 6290	. . . . . . . . 9 |- ( y = <. C , D >. -> ( <. A , B >. F y ) = ( <. A , B >. F <. C , D >. ) )
44	43	eqeq1d 2469	. . . . . . . 8 |- ( y = <. C , D >. -> ( ( <. A , B >. F y ) = z <-> ( <. A , B >. F <. C , D >. ) = z ) )
45	42, 44	imbi12d 320	. . . . . . 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 ) ) )
46		moeq 3279	. . . . . . . . . . . 12 |- E* z z = R
47	46	mosubop 4746	. . . . . . . . . . 11 |- E* z E. u E. f ( y = <. u , f >. /\ z = R )
48	47	mosubop 4746	. . . . . . . . . 10 |- E* z E. w E. v ( x = <. w , v >. /\ E. u E. f ( y = <. u , f >. /\ z = R ) )
49		anass 649	. . . . . . . . . . . . . 14 |- ( ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R ) <-> ( x = <. w , v >. /\ ( y = <. u , f >. /\ z = R ) ) )
50	49	2exbii 1645	. . . . . . . . . . . . 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 ) ) )
51		19.42vv 1951	. . . . . . . . . . . . 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 ) ) )
52	50, 51	bitri 249	. . . . . . . . . . . 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 ) ) )
53	52	2exbii 1645	. . . . . . . . . . 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 ) ) )
54	53	mobii 2301	. . . . . . . . . 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 ) ) )
55	48, 54	mpbir 209	. . . . . . . . 9 |- E* z E. w E. v E. u E. f ( ( x = <. w , v >. /\ y = <. u , f >. ) /\ z = R )
56	55	a1i 11	. . . . . . . 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 ) )
57	56, 5	ovidi 6403	. . . . . . 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 ) )
58	16, 17, 18, 25, 31, 38, 45, 57	vtocl2gaf 3178	. . . . . 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 ) )
59	14, 15, 58	syl2an 477	. . . . 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 ) )
60	13, 59	sylbird 235	. . . 4 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( z = S -> ( <. A , B >. F <. C , D >. ) = z ) )
61		eqeq2 2482	. . . 4 |- ( z = S -> ( ( <. A , B >. F <. C , D >. ) = z <-> ( <. A , B >. F <. C , D >. ) = S ) )
62	60, 61	mpbidi 216	. . 3 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( z = S -> ( <. A , B >. F <. C , D >. ) = S ) )
63	3, 10, 62	exlimd 1861	. 2 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( E. z z = S -> ( <. A , B >. F <. C , D >. ) = S ) )
64	2, 63	mpi 17	1 |- ( ( ( A e. H /\ B e. H ) /\ ( C e. H /\ D e. H ) ) -> ( <. A , B >. F <. C , D >. ) = S )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1379   E.wex 1596   e. wcel 1767   E*wmo 2276   _Vcvv 3113   <.cop 4033   X. cxp 4997  (class class class)co 6282   {coprab 6283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286

This theorem is referenced by:  addcnsr  9508  mulcnsr  9509