Theorem dibf11N 35958

Description: The partial isomorphism A for a lattice K is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dibcl.h	|- H = ( LHyp ` K )
dibcl.i	|- I = ( ( DIsoB ` K ) ` W )

Assertion

Ref	Expression
dibf11N	|- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )

Proof of Theorem dibf11N

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2467	. . . 4 |- ( Base ` K ) = ( Base ` K )
2		eqid 2467	. . . 4 |- ( le ` K ) = ( le ` K )
3		dibcl.h	. . . 4 |- H = ( LHyp ` K )
4		dibcl.i	. . . 4 |- I = ( ( DIsoB ` K ) ` W )
5	1, 2, 3, 4	dibfnN 35953	. . 3 |- ( ( K e. HL /\ W e. H ) -> I Fn { x e. ( Base ` K ) | x ( le ` K ) W } )
6		fnfun 5676	. . . 4 |- ( I Fn { x e. ( Base ` K ) | x ( le ` K ) W } -> Fun I )
7		funfn 5615	. . . 4 |- ( Fun I <-> I Fn dom I )
8	6, 7	sylib 196	. . 3 |- ( I Fn { x e. ( Base ` K ) | x ( le ` K ) W } -> I Fn dom I )
9	5, 8	syl 16	. 2 |- ( ( K e. HL /\ W e. H ) -> I Fn dom I )
10		eqidd 2468	. 2 |- ( ( K e. HL /\ W e. H ) -> ran I = ran I )
11	1, 2, 3, 4	dibeldmN 35955	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( x e. dom I <-> ( x e. ( Base ` K ) /\ x ( le ` K ) W ) ) )
12	1, 2, 3, 4	dibeldmN 35955	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( y e. dom I <-> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) )
13	11, 12	anbi12d 710	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( ( x e. dom I /\ y e. dom I ) <-> ( ( x e. ( Base ` K ) /\ x ( le ` K ) W ) /\ ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) ) )
14	1, 2, 3, 4	dib11N 35957	. . . . . 6 |- ( ( ( K e. HL /\ W e. H ) /\ ( x e. ( Base ` K ) /\ x ( le ` K ) W ) /\ ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) -> ( ( I ` x ) = ( I ` y ) <-> x = y ) )
15	14	biimpd 207	. . . . 5 |- ( ( ( K e. HL /\ W e. H ) /\ ( x e. ( Base ` K ) /\ x ( le ` K ) W ) /\ ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) -> ( ( I ` x ) = ( I ` y ) -> x = y ) )
16	15	3expib 1199	. . . 4 |- ( ( K e. HL /\ W e. H ) -> ( ( ( x e. ( Base ` K ) /\ x ( le ` K ) W ) /\ ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) -> ( ( I ` x ) = ( I ` y ) -> x = y ) ) )
17	13, 16	sylbid 215	. . 3 |- ( ( K e. HL /\ W e. H ) -> ( ( x e. dom I /\ y e. dom I ) -> ( ( I ` x ) = ( I ` y ) -> x = y ) ) )
18	17	ralrimivv 2884	. 2 |- ( ( K e. HL /\ W e. H ) -> A. x e. dom I A. y e. dom I ( ( I ` x ) = ( I ` y ) -> x = y ) )
19		dff1o6 6167	. 2 |- ( I : dom I -1-1-onto-> ran I <-> ( I Fn dom I /\ ran I = ran I /\ A. x e. dom I A. y e. dom I ( ( I ` x ) = ( I ` y ) -> x = y ) ) )
20	9, 10, 18, 19	syl3anbrc 1180	1 |- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 973   = wceq 1379   e. wcel 1767   A.wral 2814   {crab 2818   class class class wbr 4447   dom cdm 4999   ran crn 5000   Fun wfun 5580   Fn wfn 5581   -1-1-onto->wf1o 5585   ` cfv 5586   Basecbs 14486   lecple 14558   HLchlt 34147   LHypclh 34780   DIsoBcdib 35935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-riotaBAD 33756

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-undef 6999  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-llines 34294  df-lplanes 34295  df-lvols 34296  df-lines 34297  df-psubsp 34299  df-pmap 34300  df-padd 34592  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-disoa 35826  df-dib 35936

This theorem is referenced by:  dibintclN  35964