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Theorem diaf11N 31312
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 2285 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2285 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 dia1o.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dia1o.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diafn 31297 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
6 fnfun 5343 . . . 4  |-  ( I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  ->  Fun  I )
7 funfn 5285 . . . 4  |-  ( Fun  I  <->  I  Fn  dom  I )
86, 7sylib 188 . . 3  |-  ( I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  ->  I  Fn  dom  I )
95, 8syl 15 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  dom  I
)
10 eqidd 2286 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ran  I  =  ran  I )
111, 2, 3, 4diaeldm 31299 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( x  e.  dom  I 
<->  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) ) )
121, 2, 3, 4diaeldm 31299 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( y  e.  dom  I 
<->  ( y  e.  (
Base `  K )  /\  y ( le `  K ) W ) ) )
1311, 12anbi12d 691 . . . 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 31311 . . . . . 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 198 . . . . 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 1154 . . . 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 206 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( ( x  e. 
dom  I  /\  y  e.  dom  I )  -> 
( ( I `  x )  =  ( I `  y )  ->  x  =  y ) ) )
1817ralrimivv 2636 . 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 5793 . 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 1136 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 358    /\ w3a 934    = wceq 1625    e. wcel 1686   A.wral 2545   {crab 2549   class class class wbr 4025   dom cdm 4691   ran crn 4692   Fun wfun 5251    Fn wfn 5252   -1-1-onto->wf1o 5256   ` cfv 5257   Basecbs 13150   lecple 13217   HLchlt 29613   LHypclh 30246   DIsoAcdia 31291
This theorem is referenced by:  diaclN  31313  diacnvclN  31314  dia1elN  31317  diainN  31320  diaintclN  31321  diasslssN  31322  docaclN  31387  diaocN  31388  doca3N  31390  diaf1oN  31393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4311  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-1st 6124  df-2nd 6125  df-undef 6300  df-riota 6306  df-map 6776  df-poset 14082  df-plt 14094  df-lub 14110  df-glb 14111  df-join 14112  df-meet 14113  df-p0 14147  df-p1 14148  df-lat 14154  df-clat 14216  df-oposet 29439  df-ol 29441  df-oml 29442  df-covers 29529  df-ats 29530  df-atl 29561  df-cvlat 29585  df-hlat 29614  df-llines 29760  df-lplanes 29761  df-lvols 29762  df-lines 29763  df-psubsp 29765  df-pmap 29766  df-padd 30058  df-lhyp 30250  df-laut 30251  df-ldil 30366  df-ltrn 30367  df-trl 30421  df-disoa 31292
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