Metamath Proof Explorer

Theorem dibopelval2

Description: Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014) (Revised by Mario Carneiro, 6-May-2015)

Ref Expression
Hypotheses dibval2.b 
|- B = ( Base ` K )
dibval2.l 
|- .<_ = ( le ` K )
dibval2.h 
|- H = ( LHyp ` K )
dibval2.t 
|- T = ( ( LTrn ` K ) ` W )
dibval2.o 
|- .0. = ( f e. T |-> ( _I |` B ) )
dibval2.j 
|- J = ( ( DIsoA ` K ) ` W )
dibval2.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibopelval2 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( F e. ( J ` X ) /\ S = .0. ) ) )

Proof

Step Hyp Ref Expression
1 dibval2.b 
 |-  B = ( Base ` K )
2 dibval2.l 
 |-  .<_ = ( le ` K )
3 dibval2.h 
 |-  H = ( LHyp ` K )
4 dibval2.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 dibval2.o 
 |-  .0. = ( f e. T |-> ( _I |` B ) )
6 dibval2.j 
 |-  J = ( ( DIsoA ` K ) ` W )
7 dibval2.i 
 |-  I = ( ( DIsoB ` K ) ` W )
8 1 2 3 4 5 6 7 dibval2 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( J ` X ) X. { .0. } ) )
9 8 eleq2d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> <. F , S >. e. ( ( J ` X ) X. { .0. } ) ) )
10 opelxp 
 |-  ( <. F , S >. e. ( ( J ` X ) X. { .0. } ) <-> ( F e. ( J ` X ) /\ S e. { .0. } ) )
11 4 fvexi 
 |-  T e. _V
12 11 mptex 
 |-  ( f e. T |-> ( _I |` B ) ) e. _V
13 5 12 eqeltri 
 |-  .0. e. _V
14 13 elsn2 
 |-  ( S e. { .0. } <-> S = .0. )
15 14 anbi2i 
 |-  ( ( F e. ( J ` X ) /\ S e. { .0. } ) <-> ( F e. ( J ` X ) /\ S = .0. ) )
16 10 15 bitri 
 |-  ( <. F , S >. e. ( ( J ` X ) X. { .0. } ) <-> ( F e. ( J ` X ) /\ S = .0. ) )
17 9 16 bitrdi 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( <. F , S >. e. ( I ` X ) <-> ( F e. ( J ` X ) /\ S = .0. ) ) )