Step |
Hyp |
Ref |
Expression |
1 |
|
dicfn.l |
|- .<_ = ( le ` K ) |
2 |
|
dicfn.a |
|- A = ( Atoms ` K ) |
3 |
|
dicfn.h |
|- H = ( LHyp ` K ) |
4 |
|
dicfn.i |
|- I = ( ( DIsoC ` K ) ` W ) |
5 |
|
breq1 |
|- ( p = q -> ( p .<_ W <-> q .<_ W ) ) |
6 |
5
|
notbid |
|- ( p = q -> ( -. p .<_ W <-> -. q .<_ W ) ) |
7 |
6
|
elrab |
|- ( q e. { p e. A | -. p .<_ W } <-> ( q e. A /\ -. q .<_ W ) ) |
8 |
|
eqid |
|- ( ( oc ` K ) ` W ) = ( ( oc ` K ) ` W ) |
9 |
|
eqid |
|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W ) |
10 |
|
eqid |
|- ( ( TEndo ` K ) ` W ) = ( ( TEndo ` K ) ` W ) |
11 |
1 2 3 8 9 10 4
|
dicval |
|- ( ( ( K e. V /\ W e. H ) /\ ( q e. A /\ -. q .<_ W ) ) -> ( I ` q ) = { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) |
12 |
|
fvex |
|- ( I ` q ) e. _V |
13 |
11 12
|
eqeltrrdi |
|- ( ( ( K e. V /\ W e. H ) /\ ( q e. A /\ -. q .<_ W ) ) -> { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } e. _V ) |
14 |
7 13
|
sylan2b |
|- ( ( ( K e. V /\ W e. H ) /\ q e. { p e. A | -. p .<_ W } ) -> { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } e. _V ) |
15 |
14
|
ralrimiva |
|- ( ( K e. V /\ W e. H ) -> A. q e. { p e. A | -. p .<_ W } { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } e. _V ) |
16 |
|
eqid |
|- ( q e. { p e. A | -. p .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) = ( q e. { p e. A | -. p .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) |
17 |
16
|
fnmpt |
|- ( A. q e. { p e. A | -. p .<_ W } { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } e. _V -> ( q e. { p e. A | -. p .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) Fn { p e. A | -. p .<_ W } ) |
18 |
15 17
|
syl |
|- ( ( K e. V /\ W e. H ) -> ( q e. { p e. A | -. p .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) Fn { p e. A | -. p .<_ W } ) |
19 |
1 2 3 8 9 10 4
|
dicfval |
|- ( ( K e. V /\ W e. H ) -> I = ( q e. { p e. A | -. p .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) ) |
20 |
19
|
fneq1d |
|- ( ( K e. V /\ W e. H ) -> ( I Fn { p e. A | -. p .<_ W } <-> ( q e. { p e. A | -. p .<_ W } |-> { <. f , s >. | ( f = ( s ` ( iota_ u e. ( ( LTrn ` K ) ` W ) ( u ` ( ( oc ` K ) ` W ) ) = q ) ) /\ s e. ( ( TEndo ` K ) ` W ) ) } ) Fn { p e. A | -. p .<_ W } ) ) |
21 |
18 20
|
mpbird |
|- ( ( K e. V /\ W e. H ) -> I Fn { p e. A | -. p .<_ W } ) |