Metamath Proof Explorer


Theorem dibopelvalN

Description: Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014) (Revised by Mario Carneiro, 6-May-2015) (New usage is discouraged.)

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

Proof

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