Theorem ov6g 6798

Description: The value of an operation class abstraction. Special case. (Contributed by NM, 13-Nov-2006.)

Hypotheses

Ref	Expression
ov6g.1	|- ( <. x , y >. = <. A , B >. -> R = S )
ov6g.2	|- F = { <. <. x , y >. , z >. | ( <. x , y >. e. C /\ z = R ) }

Assertion

Ref	Expression
ov6g	|- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( A F B ) = S )

Distinct variable groups:   x, y, z, A   x, B, y, z   x, C, y, z   z, R   x, S, y, z

Allowed substitution hints:   R( x, y)   F( x, y, z)   G( x, y, z)   H( x, y, z)   J( x, y, z)

Proof of Theorem ov6g

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 6653	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		eqid 2622	. . . . . 6 |- S = S
3		biidd 252	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( S = S <-> S = S ) )
4	3	copsex2g 4958	. . . . . 6 |- ( ( A e. G /\ B e. H ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) <-> S = S ) )
5	2, 4	mpbiri 248	. . . . 5 |- ( ( A e. G /\ B e. H ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) )
6	5	3adant3 1081	. . . 4 |- ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) )
7	6	adantr 481	. . 3 |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) )
8		eqeq1 2626	. . . . . . . 8 |- ( w = <. A , B >. -> ( w = <. x , y >. <-> <. A , B >. = <. x , y >. ) )
9	8	anbi1d 741	. . . . . . 7 |- ( w = <. A , B >. -> ( ( w = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = R ) ) )
10		ov6g.1	. . . . . . . . . 10 |- ( <. x , y >. = <. A , B >. -> R = S )
11	10	eqeq2d 2632	. . . . . . . . 9 |- ( <. x , y >. = <. A , B >. -> ( z = R <-> z = S ) )
12	11	eqcoms 2630	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> ( z = R <-> z = S ) )
13	12	pm5.32i 669	. . . . . . 7 |- ( ( <. A , B >. = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = S ) )
14	9, 13	syl6bb 276	. . . . . 6 |- ( w = <. A , B >. -> ( ( w = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = S ) ) )
15	14	2exbidv 1852	. . . . 5 |- ( w = <. A , B >. -> ( E. x E. y ( w = <. x , y >. /\ z = R ) <-> E. x E. y ( <. A , B >. = <. x , y >. /\ z = S ) ) )
16		eqeq1 2626	. . . . . . 7 |- ( z = S -> ( z = S <-> S = S ) )
17	16	anbi2d 740	. . . . . 6 |- ( z = S -> ( ( <. A , B >. = <. x , y >. /\ z = S ) <-> ( <. A , B >. = <. x , y >. /\ S = S ) ) )
18	17	2exbidv 1852	. . . . 5 |- ( z = S -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ z = S ) <-> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) ) )
19		moeq 3382	. . . . . . 7 |- E* z z = R
20	19	mosubop 4973	. . . . . 6 |- E* z E. x E. y ( w = <. x , y >. /\ z = R )
21	20	a1i 11	. . . . 5 |- ( w e. C -> E* z E. x E. y ( w = <. x , y >. /\ z = R ) )
22		ov6g.2	. . . . . 6 |- F = { <. <. x , y >. , z >. | ( <. x , y >. e. C /\ z = R ) }
23		dfoprab2 6701	. . . . . 6 |- { <. <. x , y >. , z >. | ( <. x , y >. e. C /\ z = R ) } = { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) }
24		eleq1 2689	. . . . . . . . . . . 12 |- ( w = <. x , y >. -> ( w e. C <-> <. x , y >. e. C ) )
25	24	anbi1d 741	. . . . . . . . . . 11 |- ( w = <. x , y >. -> ( ( w e. C /\ z = R ) <-> ( <. x , y >. e. C /\ z = R ) ) )
26	25	pm5.32i 669	. . . . . . . . . 10 |- ( ( w = <. x , y >. /\ ( w e. C /\ z = R ) ) <-> ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) )
27		an12 838	. . . . . . . . . 10 |- ( ( w = <. x , y >. /\ ( w e. C /\ z = R ) ) <-> ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) )
28	26, 27	bitr3i 266	. . . . . . . . 9 |- ( ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) <-> ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) )
29	28	2exbii 1775	. . . . . . . 8 |- ( E. x E. y ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) <-> E. x E. y ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) )
30		19.42vv 1920	. . . . . . . 8 |- ( E. x E. y ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) <-> ( w e. C /\ E. x E. y ( w = <. x , y >. /\ z = R ) ) )
31	29, 30	bitri 264	. . . . . . 7 |- ( E. x E. y ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) <-> ( w e. C /\ E. x E. y ( w = <. x , y >. /\ z = R ) ) )
32	31	opabbii 4717	. . . . . 6 |- { <. w , z >. | E. x E. y ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) } = { <. w , z >. | ( w e. C /\ E. x E. y ( w = <. x , y >. /\ z = R ) ) }
33	22, 23, 32	3eqtri 2648	. . . . 5 |- F = { <. w , z >. | ( w e. C /\ E. x E. y ( w = <. x , y >. /\ z = R ) ) }
34	15, 18, 21, 33	fvopab3ig 6278	. . . 4 |- ( ( <. A , B >. e. C /\ S e. J ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) -> ( F ` <. A , B >. ) = S ) )
35	34	3ad2antl3 1225	. . 3 |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) -> ( F ` <. A , B >. ) = S ) )
36	7, 35	mpd 15	. 2 |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( F ` <. A , B >. ) = S )
37	1, 36	syl5eq 2668	1 |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   E*wmo 2471   <.cop 4183   {copab 4712   ` cfv 5888  (class class class)co 6650   {coprab 6651

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654

This theorem is referenced by: (None)