Theorem ov6g 5658

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 5535	. 2 |- ( A F B ) = ( F ` <. A , B >. )
2		eqid 2081	. . . . . 6 |- S = S
3		biidd 170	. . . . . . 7 |- ( ( x = A /\ y = B ) -> ( S = S <-> S = S ) )
4	3	copsex2g 4001	. . . . . 6 |- ( ( A e. G /\ B e. H ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) <-> S = S ) )
5	2, 4	mpbiri 166	. . . . 5 |- ( ( A e. G /\ B e. H ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) )
6	5	3adant3 958	. . . 4 |- ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) )
7	6	adantr 270	. . 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 2087	. . . . . . . 8 |- ( w = <. A , B >. -> ( w = <. x , y >. <-> <. A , B >. = <. x , y >. ) )
9	8	anbi1d 452	. . . . . . 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 2092	. . . . . . . . 9 |- ( <. x , y >. = <. A , B >. -> ( z = R <-> z = S ) )
12	11	eqcoms 2084	. . . . . . . 8 |- ( <. A , B >. = <. x , y >. -> ( z = R <-> z = S ) )
13	12	pm5.32i 441	. . . . . . 7 |- ( ( <. A , B >. = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = S ) )
14	9, 13	syl6bb 194	. . . . . 6 |- ( w = <. A , B >. -> ( ( w = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = S ) ) )
15	14	2exbidv 1789	. . . . 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 2087	. . . . . . 7 |- ( z = S -> ( z = S <-> S = S ) )
17	16	anbi2d 451	. . . . . 6 |- ( z = S -> ( ( <. A , B >. = <. x , y >. /\ z = S ) <-> ( <. A , B >. = <. x , y >. /\ S = S ) ) )
18	17	2exbidv 1789	. . . . 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 2767	. . . . . . 7 |- E* z z = R
20	19	mosubop 4424	. . . . . 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 5572	. . . . . 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 2141	. . . . . . . . . . . 12 |- ( w = <. x , y >. -> ( w e. C <-> <. x , y >. e. C ) )
25	24	anbi1d 452	. . . . . . . . . . 11 |- ( w = <. x , y >. -> ( ( w e. C /\ z = R ) <-> ( <. x , y >. e. C /\ z = R ) ) )
26	25	pm5.32i 441	. . . . . . . . . 10 |- ( ( w = <. x , y >. /\ ( w e. C /\ z = R ) ) <-> ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) )
27		an12 525	. . . . . . . . . 10 |- ( ( w = <. x , y >. /\ ( w e. C /\ z = R ) ) <-> ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) )
28	26, 27	bitr3i 184	. . . . . . . . 9 |- ( ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) <-> ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) )
29	28	2exbii 1537	. . . . . . . 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 1829	. . . . . . . 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 182	. . . . . . 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 3845	. . . . . 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 2105	. . . . 5 |- F = { <. w , z >. | ( w e. C /\ E. x E. y ( w = <. x , y >. /\ z = R ) ) }
34	15, 18, 21, 33	fvopab3ig 5267	. . . 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 1102	. . 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	syl5eq 2125	1 |- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( A F B ) = S )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 919   = wceq 1284   E.wex 1421   e. wcel 1433   E*wmo 1942   <.cop 3401   {copab 3838   ` cfv 4922  (class class class)co 5532   {coprab 5533

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-oprab 5536

This theorem is referenced by: (None)