Theorem mapd11

Description: The map defined by df-mapd is one-to-one. Property (c) of Baer p. 40. (Contributed by NM, 12-Mar-2015)

	Ref	Expression
Hypotheses	mapdord.h	|- H = ( LHyp ` K )
	mapdord.u	|- U = ( ( DVecH ` K ) ` W )
	mapdord.s	|- S = ( LSubSp ` U )
	mapdord.m	|- M = ( ( mapd ` K ) ` W )
	mapdord.k	|- ( ph -> ( K e. HL /\ W e. H ) )
	mapdord.x	|- ( ph -> X e. S )
	mapdord.y	|- ( ph -> Y e. S )
Assertion	mapd11	|- ( ph -> ( ( M ` X ) = ( M ` Y ) <-> X = Y ) )

Proof

Step	Hyp	Ref	Expression
1		mapdord.h	|- H = ( LHyp ` K )
2		mapdord.u	|- U = ( ( DVecH ` K ) ` W )
3		mapdord.s	|- S = ( LSubSp ` U )
4		mapdord.m	|- M = ( ( mapd ` K ) ` W )
5		mapdord.k	|- ( ph -> ( K e. HL /\ W e. H ) )
6		mapdord.x	|- ( ph -> X e. S )
7		mapdord.y	|- ( ph -> Y e. S )
8	1 2 3 4 5 6 7	mapdord	|- ( ph -> ( ( M ` X ) C_ ( M ` Y ) <-> X C_ Y ) )
9	1 2 3 4 5 7 6	mapdord	|- ( ph -> ( ( M ` Y ) C_ ( M ` X ) <-> Y C_ X ) )
10	8 9	anbi12d	|- ( ph -> ( ( ( M ` X ) C_ ( M ` Y ) /\ ( M ` Y ) C_ ( M ` X ) ) <-> ( X C_ Y /\ Y C_ X ) ) )
11		eqss	|- ( ( M ` X ) = ( M ` Y ) <-> ( ( M ` X ) C_ ( M ` Y ) /\ ( M ` Y ) C_ ( M ` X ) ) )
12		eqss	|- ( X = Y <-> ( X C_ Y /\ Y C_ X ) )
13	10 11 12	3bitr4g	|- ( ph -> ( ( M ` X ) = ( M ` Y ) <-> X = Y ) )