Theorem dibf11N 35125

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 2452	. . . 4 |- ( Base ` K ) = ( Base ` K )
2		eqid 2452	. . . 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 35120	. . 3 |- ( ( K e. HL /\ W e. H ) -> I Fn { x e. ( Base ` K ) | x ( le ` K ) W } )
6		fnfun 5611	. . . 4 |- ( I Fn { x e. ( Base ` K ) | x ( le ` K ) W } -> Fun I )
7		funfn 5550	. . . 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 2453	. 2 |- ( ( K e. HL /\ W e. H ) -> ran I = ran I )
11	1, 2, 3, 4	dibeldmN 35122	. . . . 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 35122	. . . . 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 35124	. . . . . 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 1191	. . . 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 2907	. 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 6086	. 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 1172	1 |- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1758   A.wral 2796   {crab 2800   class class class wbr 4395   dom cdm 4943   ran crn 4944   Fun wfun 5515   Fn wfn 5516   -1-1-onto->wf1o 5520   ` cfv 5521   Basecbs 14287   lecple 14359   HLchlt 33314   LHypclh 33947   DIsoBcdib 35102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-riotaBAD 32923

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-iin 4277  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-1st 6682  df-2nd 6683  df-undef 6897  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-llines 33461  df-lplanes 33462  df-lvols 33463  df-lines 33464  df-psubsp 33466  df-pmap 33467  df-padd 33759  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122  df-disoa 34993  df-dib 35103

This theorem is referenced by:  dibintclN  35131