Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  diaf11N Unicode version

Theorem diaf11N 31164
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 2387 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2387 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 dia1o.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dia1o.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diafn 31149 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
6 fnfun 5482 . . . 4  |-  ( I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  ->  Fun  I )
7 funfn 5422 . . . 4  |-  ( Fun  I  <->  I  Fn  dom  I )
86, 7sylib 189 . . 3  |-  ( I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  ->  I  Fn  dom  I )
95, 8syl 16 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  dom  I
)
10 eqidd 2388 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ran  I  =  ran  I )
111, 2, 3, 4diaeldm 31151 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( x  e.  dom  I 
<->  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) ) )
121, 2, 3, 4diaeldm 31151 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( y  e.  dom  I 
<->  ( y  e.  (
Base `  K )  /\  y ( le `  K ) W ) ) )
1311, 12anbi12d 692 . . . 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 31163 . . . . . 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 199 . . . . 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 1156 . . . 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 207 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( x  e. 
dom  I  /\  y  e.  dom  I )  -> 
( ( I `  x )  =  ( I `  y )  ->  x  =  y ) ) )
1817ralrimivv 2740 . 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 5952 . 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 1138 1  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I : dom  I -1-1-onto-> ran  I )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2649   {crab 2653   class class class wbr 4153   dom cdm 4818   ran crn 4819   Fun wfun 5388    Fn wfn 5389   -1-1-onto->wf1o 5393   ` cfv 5394   Basecbs 13396   lecple 13463   HLchlt 29465   LHypclh 30098   DIsoAcdia 31143
This theorem is referenced by:  diaclN  31165  diacnvclN  31166  dia1elN  31169  diainN  31172  diaintclN  31173  diasslssN  31174  docaclN  31239  diaocN  31240  doca3N  31242  diaf1oN  31245
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-undef 6479  df-riota 6485  df-map 6956  df-poset 14330  df-plt 14342  df-lub 14358  df-glb 14359  df-join 14360  df-meet 14361  df-p0 14395  df-p1 14396  df-lat 14402  df-clat 14464  df-oposet 29291  df-ol 29293  df-oml 29294  df-covers 29381  df-ats 29382  df-atl 29413  df-cvlat 29437  df-hlat 29466  df-llines 29612  df-lplanes 29613  df-lvols 29614  df-lines 29615  df-psubsp 29617  df-pmap 29618  df-padd 29910  df-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-disoa 31144
  Copyright terms: Public domain W3C validator