Theorem dibf11N 34161

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 2402	. . . 4 |- ( Base ` K ) = ( Base ` K )
2		eqid 2402	. . . 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 34156	. . 3 |- ( ( K e. HL /\ W e. H ) -> I Fn { x e. ( Base ` K ) | x ( le ` K ) W } )
6		fnfun 5658	. . . 4 |- ( I Fn { x e. ( Base ` K ) | x ( le ` K ) W } -> Fun I )
7		funfn 5597	. . . 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 17	. 2 |- ( ( K e. HL /\ W e. H ) -> I Fn dom I )
10		eqidd 2403	. 2 |- ( ( K e. HL /\ W e. H ) -> ran I = ran I )
11	1, 2, 3, 4	dibeldmN 34158	. . . . 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 34158	. . . . 5 |- ( ( K e. HL /\ W e. H ) -> ( y e. dom I <-> ( y e. ( Base ` K ) /\ y ( le ` K ) W ) ) )
13	11, 12	anbi12d 709	. . . 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 34160	. . . . . 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 1200	. . . 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 2823	. 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 6161	. 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 1181	1 |- ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )

Syntax hints:   -> wi 4   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   A.wral 2753   {crab 2757   class class class wbr 4394   dom cdm 4822   ran crn 4823   Fun wfun 5562   Fn wfn 5563   -1-1-onto->wf1o 5567   ` cfv 5568   Basecbs 14839   lecple 14914   HLchlt 32348   LHypclh 32981   DIsoBcdib 34138

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-riotaBAD 31957

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-1st 6783  df-2nd 6784  df-undef 7004  df-map 7458  df-preset 15879  df-poset 15897  df-plt 15910  df-lub 15926  df-glb 15927  df-join 15928  df-meet 15929  df-p0 15991  df-p1 15992  df-lat 15998  df-clat 16060  df-oposet 32174  df-ol 32176  df-oml 32177  df-covers 32264  df-ats 32265  df-atl 32296  df-cvlat 32320  df-hlat 32349  df-llines 32495  df-lplanes 32496  df-lvols 32497  df-lines 32498  df-psubsp 32500  df-pmap 32501  df-padd 32793  df-lhyp 32985  df-laut 32986  df-ldil 33101  df-ltrn 33102  df-trl 33157  df-disoa 34029  df-dib 34139

This theorem is referenced by:  dibintclN  34167