Theorem diaf1oN

Description: The partial isomorphism A for a lattice K is a one-to-one, onto function. Part of Lemma M of Crawley p. 121 line 12, with closed subspaces rather than subspaces. See diadm for the domain. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvadia.h	|- H = ( LHyp ` K )
	dvadia.u	|- U = ( ( DVecA ` K ) ` W )
	dvadia.i	|- I = ( ( DIsoA ` K ) ` W )
	dvadia.n	|- ._|_ = ( ( ocA ` K ) ` W )
	dvadia.s	|- S = ( LSubSp ` U )
Assertion	diaf1oN	|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )

Proof

Step	Hyp	Ref	Expression
1		dvadia.h	|- H = ( LHyp ` K )
2		dvadia.u	|- U = ( ( DVecA ` K ) ` W )
3		dvadia.i	|- I = ( ( DIsoA ` K ) ` W )
4		dvadia.n	|- ._|_ = ( ( ocA ` K ) ` W )
5		dvadia.s	|- S = ( LSubSp ` U )
6	1 3	diaf11N	|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )
7		f1of1	|- ( I : dom I -1-1-onto-> ran I -> I : dom I -1-1-> ran I )
8	6 7	syl	|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-> ran I )
9	1 2 3 4 5	diarnN	|- ( ( K e. HL /\ W e. H ) -> ran I = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )
10		f1eq3	|- ( ran I = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } -> ( I : dom I -1-1-> ran I <-> I : dom I -1-1-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } ) )
11	9 10	syl	|- ( ( K e. HL /\ W e. H ) -> ( I : dom I -1-1-> ran I <-> I : dom I -1-1-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } ) )
12	8 11	mpbid	|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )
13		dff1o5	|- ( I : dom I -1-1-onto-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } <-> ( I : dom I -1-1-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } /\ ran I = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } ) )
14	12 9 13	sylanbrc	|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )