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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.
StepHypRef 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 )
51, 2, 3, 4dibfnN 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 )
86, 7sylib 196 . . 3  |-  ( I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  ->  I  Fn  dom  I )
95, 8syl 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 )
111, 2, 3, 4dibeldmN 34158 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( x  e.  dom  I 
<->  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) ) )
121, 2, 3, 4dibeldmN 34158 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( y  e.  dom  I 
<->  ( y  e.  (
Base `  K )  /\  y ( le `  K ) W ) ) )
1311, 12anbi12d 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 ) ) ) )
141, 2, 3, 4dib11N 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 ) )
1514biimpd 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 )
)
16153expib 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 )
) )
1713, 16sylbid 215 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( x  e. 
dom  I  /\  y  e.  dom  I )  -> 
( ( I `  x )  =  ( I `  y )  ->  x  =  y ) ) )
1817ralrimivv 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 ) ) )
209, 10, 18, 19syl3anbrc 1181 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
Colors of variables: wff setvar class
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
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