Step |
Hyp |
Ref |
Expression |
1 |
|
mgcoval.1 |
|- A = ( Base ` V ) |
2 |
|
mgcoval.2 |
|- B = ( Base ` W ) |
3 |
|
mgcoval.3 |
|- .<_ = ( le ` V ) |
4 |
|
mgcoval.4 |
|- .c_ = ( le ` W ) |
5 |
|
mgcval.1 |
|- H = ( V MGalConn W ) |
6 |
|
mgcval.2 |
|- ( ph -> V e. Proset ) |
7 |
|
mgcval.3 |
|- ( ph -> W e. Proset ) |
8 |
|
mgccole.1 |
|- ( ph -> F H G ) |
9 |
|
mgccole1.2 |
|- ( ph -> X e. A ) |
10 |
1 2 3 4 5 6 7
|
mgcval |
|- ( ph -> ( F H G <-> ( ( F : A --> B /\ G : B --> A ) /\ A. x e. A A. y e. B ( ( F ` x ) .c_ y <-> x .<_ ( G ` y ) ) ) ) ) |
11 |
8 10
|
mpbid |
|- ( ph -> ( ( F : A --> B /\ G : B --> A ) /\ A. x e. A A. y e. B ( ( F ` x ) .c_ y <-> x .<_ ( G ` y ) ) ) ) |
12 |
11
|
simplld |
|- ( ph -> F : A --> B ) |
13 |
12 9
|
ffvelrnd |
|- ( ph -> ( F ` X ) e. B ) |
14 |
2 4
|
prsref |
|- ( ( W e. Proset /\ ( F ` X ) e. B ) -> ( F ` X ) .c_ ( F ` X ) ) |
15 |
7 13 14
|
syl2anc |
|- ( ph -> ( F ` X ) .c_ ( F ` X ) ) |
16 |
|
fveq2 |
|- ( x = X -> ( F ` x ) = ( F ` X ) ) |
17 |
16
|
breq1d |
|- ( x = X -> ( ( F ` x ) .c_ y <-> ( F ` X ) .c_ y ) ) |
18 |
|
breq1 |
|- ( x = X -> ( x .<_ ( G ` y ) <-> X .<_ ( G ` y ) ) ) |
19 |
17 18
|
bibi12d |
|- ( x = X -> ( ( ( F ` x ) .c_ y <-> x .<_ ( G ` y ) ) <-> ( ( F ` X ) .c_ y <-> X .<_ ( G ` y ) ) ) ) |
20 |
19
|
ralbidv |
|- ( x = X -> ( A. y e. B ( ( F ` x ) .c_ y <-> x .<_ ( G ` y ) ) <-> A. y e. B ( ( F ` X ) .c_ y <-> X .<_ ( G ` y ) ) ) ) |
21 |
11
|
simprd |
|- ( ph -> A. x e. A A. y e. B ( ( F ` x ) .c_ y <-> x .<_ ( G ` y ) ) ) |
22 |
20 21 9
|
rspcdva |
|- ( ph -> A. y e. B ( ( F ` X ) .c_ y <-> X .<_ ( G ` y ) ) ) |
23 |
|
simpr |
|- ( ( ph /\ y = ( F ` X ) ) -> y = ( F ` X ) ) |
24 |
23
|
breq2d |
|- ( ( ph /\ y = ( F ` X ) ) -> ( ( F ` X ) .c_ y <-> ( F ` X ) .c_ ( F ` X ) ) ) |
25 |
23
|
fveq2d |
|- ( ( ph /\ y = ( F ` X ) ) -> ( G ` y ) = ( G ` ( F ` X ) ) ) |
26 |
25
|
breq2d |
|- ( ( ph /\ y = ( F ` X ) ) -> ( X .<_ ( G ` y ) <-> X .<_ ( G ` ( F ` X ) ) ) ) |
27 |
24 26
|
bibi12d |
|- ( ( ph /\ y = ( F ` X ) ) -> ( ( ( F ` X ) .c_ y <-> X .<_ ( G ` y ) ) <-> ( ( F ` X ) .c_ ( F ` X ) <-> X .<_ ( G ` ( F ` X ) ) ) ) ) |
28 |
13 27
|
rspcdv |
|- ( ph -> ( A. y e. B ( ( F ` X ) .c_ y <-> X .<_ ( G ` y ) ) -> ( ( F ` X ) .c_ ( F ` X ) <-> X .<_ ( G ` ( F ` X ) ) ) ) ) |
29 |
22 28
|
mpd |
|- ( ph -> ( ( F ` X ) .c_ ( F ` X ) <-> X .<_ ( G ` ( F ` X ) ) ) ) |
30 |
15 29
|
mpbid |
|- ( ph -> X .<_ ( G ` ( F ` X ) ) ) |