Theorem ov6g 6159

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 6020	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		eqid 2231	. . . . . 6 |- S = S
3		biidd 172	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( S = S <-> S = S ) )
4	3	copsex2g 4338	. . . . . 6 |- ( ( A e. G /\ B e. H ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) <-> S = S ) )
5	2, 4	mpbiri 168	. . . . 5 |- ( ( A e. G /\ B e. H ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) )
6	5	3adant3 1043	. . . 4 |- ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) )
7	6	adantr 276	. . 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 2238	. . . . . . . 8 |- ( w = <. A , B >. -> ( w = <. x , y >. <-> <. A , B >. = <. x , y >. ) )
9	8	anbi1d 465	. . . . . . 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 2243	. . . . . . . . 9 |- ( <. x , y >. = <. A , B >. -> ( z = R <-> z = S ) )
12	11	eqcoms 2234	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> ( z = R <-> z = S ) )
13	12	pm5.32i 454	. . . . . . 7 |- ( ( <. A , B >. = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = S ) )
14	9, 13	bitrdi 196	. . . . . 6 |- ( w = <. A , B >. -> ( ( w = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = S ) ) )
15	14	2exbidv 1916	. . . . 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 2238	. . . . . . 7 |- ( z = S -> ( z = S <-> S = S ) )
17	16	anbi2d 464	. . . . . 6 |- ( z = S -> ( ( <. A , B >. = <. x , y >. /\ z = S ) <-> ( <. A , B >. = <. x , y >. /\ S = S ) ) )
18	17	2exbidv 1916	. . . . 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 2981	. . . . . . 7 |- E* z z = R
20	19	mosubop 4792	. . . . . 6 |- E* z E. x E. y ( w = <. x , y >. /\ z = R )
21	20	a1i 9	. . . . 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 6067	. . . . . 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 2294	. . . . . . . . . . . 12 |- ( w = <. x , y >. -> ( w e. C <-> <. x , y >. e. C ) )
25	24	anbi1d 465	. . . . . . . . . . 11 |- ( w = <. x , y >. -> ( ( w e. C /\ z = R ) <-> ( <. x , y >. e. C /\ z = R ) ) )
26	25	pm5.32i 454	. . . . . . . . . 10 |- ( ( w = <. x , y >. /\ ( w e. C /\ z = R ) ) <-> ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) )
27		an12 563	. . . . . . . . . 10 |- ( ( w = <. x , y >. /\ ( w e. C /\ z = R ) ) <-> ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) )
28	26, 27	bitr3i 186	. . . . . . . . 9 |- ( ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) <-> ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) )
29	28	2exbii 1654	. . . . . . . 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 1960	. . . . . . . 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 184	. . . . . . 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 4156	. . . . . 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 2256	. . . . 5 |- F = { <. w , z >. | ( w e. C /\ E. x E. y ( w = <. x , y >. /\ z = R ) ) }
34	15, 18, 21, 33	fvopab3ig 5720	. . . 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 1187	. . 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 13	. 2 |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( F ` <. A , B >. ) = S )
37	1, 36	eqtrid 2276	1 |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1004   = wceq 1397   E.wex 1540   E*wmo 2080   e. wcel 2202   <.cop 3672   {copab 4149   ` cfv 5326  (class class class)co 6017   {coprab 6018

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-ov 6020  df-oprab 6021

This theorem is referenced by: (None)