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Theorem diaf11N 34082
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
dia1o.h  |-  H  =  ( LHyp `  K
)
dia1o.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diaf11N  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )

Proof of Theorem diaf11N
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2404 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 dia1o.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dia1o.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diafn 34067 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
6 fnfun 5661 . . . 4  |-  ( I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  ->  Fun  I )
7 funfn 5600 . . . 4  |-  ( Fun  I  <->  I  Fn  dom  I )
86, 7sylib 198 . . 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 2405 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ran  I  =  ran  I )
111, 2, 3, 4diaeldm 34069 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( x  e.  dom  I 
<->  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) ) )
121, 2, 3, 4diaeldm 34069 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( y  e.  dom  I 
<->  ( y  e.  (
Base `  K )  /\  y ( le `  K ) W ) ) )
1311, 12anbi12d 711 . . . 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, 4dia11N 34081 . . . . . 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 209 . . . . 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 1202 . . . 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 217 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( x  e. 
dom  I  /\  y  e.  dom  I )  -> 
( ( I `  x )  =  ( I `  y )  ->  x  =  y ) ) )
1817ralrimivv 2826 . 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 6164 . 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 1183 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 369    /\ w3a 976    = wceq 1407    e. wcel 1844   A.wral 2756   {crab 2760   class class class wbr 4397   dom cdm 4825   ran crn 4826   Fun wfun 5565    Fn wfn 5566   -1-1-onto->wf1o 5570   ` cfv 5571   Basecbs 14843   lecple 14918   HLchlt 32381   LHypclh 33014   DIsoAcdia 34061
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-rep 4509  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-riotaBAD 31990
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-iin 4276  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-riota 6242  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-undef 7007  df-map 7461  df-preset 15883  df-poset 15901  df-plt 15914  df-lub 15930  df-glb 15931  df-join 15932  df-meet 15933  df-p0 15995  df-p1 15996  df-lat 16002  df-clat 16064  df-oposet 32207  df-ol 32209  df-oml 32210  df-covers 32297  df-ats 32298  df-atl 32329  df-cvlat 32353  df-hlat 32382  df-llines 32528  df-lplanes 32529  df-lvols 32530  df-lines 32531  df-psubsp 32533  df-pmap 32534  df-padd 32826  df-lhyp 33018  df-laut 33019  df-ldil 33134  df-ltrn 33135  df-trl 33190  df-disoa 34062
This theorem is referenced by:  diaclN  34083  diacnvclN  34084  dia1elN  34087  diainN  34090  diaintclN  34091  diasslssN  34092  docaclN  34157  diaocN  34158  doca3N  34160  diaf1oN  34163
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