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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. ) ) )