Theorem dibn0

Description: The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014)

	Ref	Expression
Hypotheses	dibn0.b	|- B = ( Base ` K )
	dibn0.l	|- .<_ = ( le ` K )
	dibn0.h	|- H = ( LHyp ` K )
	dibn0.i	|- I = ( ( DIsoB ` K ) ` W )
Assertion	dibn0	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) =/= (/) )

Proof

Step	Hyp	Ref	Expression
1		dibn0.b	|- B = ( Base ` K )
2		dibn0.l	|- .<_ = ( le ` K )
3		dibn0.h	|- H = ( LHyp ` K )
4		dibn0.i	|- I = ( ( DIsoB ` K ) ` W )
5		eqid	|- ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
6		eqid	|- ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) = ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) )
7		eqid	|- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
8	1 2 3 5 6 7 4	dibval2	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) )
9	1 2 3 7	dian0	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( DIsoA ` K ) ` W ) ` X ) =/= (/) )
10		fvex	|- ( ( LTrn ` K ) ` W ) e. _V
11	10	mptex	|- ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) e. _V
12	11	snnz	|- { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } =/= (/)
13	9 12	jctir	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( ( DIsoA ` K ) ` W ) ` X ) =/= (/) /\ { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } =/= (/) ) )
14		xpnz	|- ( ( ( ( ( DIsoA ` K ) ` W ) ` X ) =/= (/) /\ { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } =/= (/) ) <-> ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) =/= (/) )
15	13 14	sylib	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { ( f e. ( ( LTrn ` K ) ` W ) |-> ( _I |` B ) ) } ) =/= (/) )
16	8 15	eqnetrd	|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) =/= (/) )