Step |
Hyp |
Ref |
Expression |
1 |
|
ov6g.1 |
|- ( <. x , y >. = <. A , B >. -> R = S ) |
2 |
|
ov6g.2 |
|- F = { <. <. x , y >. , z >. | ( <. x , y >. e. C /\ z = R ) } |
3 |
|
df-ov |
|- ( A F B ) = ( F ` <. A , B >. ) |
4 |
|
eqid |
|- S = S |
5 |
|
biidd |
|- ( ( x = A /\ y = B ) -> ( S = S <-> S = S ) ) |
6 |
5
|
copsex2g |
|- ( ( A e. G /\ B e. H ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) <-> S = S ) ) |
7 |
4 6
|
mpbiri |
|- ( ( A e. G /\ B e. H ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) ) |
8 |
7
|
3adant3 |
|- ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) ) |
9 |
8
|
adantr |
|- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) ) |
10 |
|
eqeq1 |
|- ( w = <. A , B >. -> ( w = <. x , y >. <-> <. A , B >. = <. x , y >. ) ) |
11 |
10
|
anbi1d |
|- ( w = <. A , B >. -> ( ( w = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = R ) ) ) |
12 |
1
|
eqeq2d |
|- ( <. x , y >. = <. A , B >. -> ( z = R <-> z = S ) ) |
13 |
12
|
eqcoms |
|- ( <. A , B >. = <. x , y >. -> ( z = R <-> z = S ) ) |
14 |
13
|
pm5.32i |
|- ( ( <. A , B >. = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = S ) ) |
15 |
11 14
|
bitrdi |
|- ( w = <. A , B >. -> ( ( w = <. x , y >. /\ z = R ) <-> ( <. A , B >. = <. x , y >. /\ z = S ) ) ) |
16 |
15
|
2exbidv |
|- ( w = <. A , B >. -> ( E. x E. y ( w = <. x , y >. /\ z = R ) <-> E. x E. y ( <. A , B >. = <. x , y >. /\ z = S ) ) ) |
17 |
|
eqeq1 |
|- ( z = S -> ( z = S <-> S = S ) ) |
18 |
17
|
anbi2d |
|- ( z = S -> ( ( <. A , B >. = <. x , y >. /\ z = S ) <-> ( <. A , B >. = <. x , y >. /\ S = S ) ) ) |
19 |
18
|
2exbidv |
|- ( z = S -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ z = S ) <-> E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) ) ) |
20 |
|
moeq |
|- E* z z = R |
21 |
20
|
mosubop |
|- E* z E. x E. y ( w = <. x , y >. /\ z = R ) |
22 |
21
|
a1i |
|- ( w e. C -> E* z E. x E. y ( w = <. x , y >. /\ z = R ) ) |
23 |
|
dfoprab2 |
|- { <. <. 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 |
|- ( w = <. x , y >. -> ( w e. C <-> <. x , y >. e. C ) ) |
25 |
24
|
anbi1d |
|- ( w = <. x , y >. -> ( ( w e. C /\ z = R ) <-> ( <. x , y >. e. C /\ z = R ) ) ) |
26 |
25
|
pm5.32i |
|- ( ( w = <. x , y >. /\ ( w e. C /\ z = R ) ) <-> ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) ) |
27 |
|
an12 |
|- ( ( w = <. x , y >. /\ ( w e. C /\ z = R ) ) <-> ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) ) |
28 |
26 27
|
bitr3i |
|- ( ( w = <. x , y >. /\ ( <. x , y >. e. C /\ z = R ) ) <-> ( w e. C /\ ( w = <. x , y >. /\ z = R ) ) ) |
29 |
28
|
2exbii |
|- ( 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 |
|- ( 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 |
|- ( 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 |
|- { <. 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 |
2 23 32
|
3eqtri |
|- F = { <. w , z >. | ( w e. C /\ E. x E. y ( w = <. x , y >. /\ z = R ) ) } |
34 |
16 19 22 33
|
fvopab3ig |
|- ( ( <. A , B >. e. C /\ S e. J ) -> ( E. x E. y ( <. A , B >. = <. x , y >. /\ S = S ) -> ( F ` <. A , B >. ) = S ) ) |
35 |
34
|
3ad2antl3 |
|- ( ( ( 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 |
9 35
|
mpd |
|- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( F ` <. A , B >. ) = S ) |
37 |
3 36
|
eqtrid |
|- ( ( ( A e. G /\ B e. H /\ <. A , B >. e. C ) /\ S e. J ) -> ( A F B ) = S ) |