Theorem cdlemeiota

Description: A translation is uniquely determined by one of its values. (Contributed by NM, 18-Apr-2013)

	Ref	Expression
Hypotheses	cdlemg1c.l	|- .<_ = ( le ` K )
	cdlemg1c.a	|- A = ( Atoms ` K )
	cdlemg1c.h	|- H = ( LHyp ` K )
	cdlemg1c.t	|- T = ( ( LTrn ` K ) ` W )
Assertion	cdlemeiota	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> F = ( iota_ f e. T ( f ` P ) = ( F ` P ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemg1c.l	|- .<_ = ( le ` K )
2		cdlemg1c.a	|- A = ( Atoms ` K )
3		cdlemg1c.h	|- H = ( LHyp ` K )
4		cdlemg1c.t	|- T = ( ( LTrn ` K ) ` W )
5		eqidd	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> ( F ` P ) = ( F ` P ) )
6		simp3	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> F e. T )
7	1 2 3 4	ltrnel	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
8	7	3com23	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) )
9	1 2 3 4	cdleme	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( ( F ` P ) e. A /\ -. ( F ` P ) .<_ W ) ) -> E! f e. T ( f ` P ) = ( F ` P ) )
10	8 9	syld3an3	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> E! f e. T ( f ` P ) = ( F ` P ) )
11		fveq1	|- ( f = F -> ( f ` P ) = ( F ` P ) )
12	11	eqeq1d	|- ( f = F -> ( ( f ` P ) = ( F ` P ) <-> ( F ` P ) = ( F ` P ) ) )
13	12	riota2	|- ( ( F e. T /\ E! f e. T ( f ` P ) = ( F ` P ) ) -> ( ( F ` P ) = ( F ` P ) <-> ( iota_ f e. T ( f ` P ) = ( F ` P ) ) = F ) )
14	6 10 13	syl2anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> ( ( F ` P ) = ( F ` P ) <-> ( iota_ f e. T ( f ` P ) = ( F ` P ) ) = F ) )
15	5 14	mpbid	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> ( iota_ f e. T ( f ` P ) = ( F ` P ) ) = F )
16	15	eqcomd	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ F e. T ) -> F = ( iota_ f e. T ( f ` P ) = ( F ` P ) ) )