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Theorem diaf11N 31536
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 2408 . . . 4  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2408 . . . 4  |-  ( le
`  K )  =  ( le `  K
)
3 dia1o.h . . . 4  |-  H  =  ( LHyp `  K
)
4 dia1o.i . . . 4  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diafn 31521 . . 3  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W } )
6 fnfun 5505 . . . 4  |-  ( I  Fn  { x  e.  ( Base `  K
)  |  x ( le `  K ) W }  ->  Fun  I )
7 funfn 5445 . . . 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 2409 . 2  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ran  I  =  ran  I )
111, 2, 3, 4diaeldm 31523 . . . . 5  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( x  e.  dom  I 
<->  ( x  e.  (
Base `  K )  /\  x ( le `  K ) W ) ) )
121, 2, 3, 4diaeldm 31523 . . . . 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 31535 . . . . . 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 2761 . 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 5976 . 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 1721   A.wral 2670   {crab 2674   class class class wbr 4176   dom cdm 4841   ran crn 4842   Fun wfun 5411    Fn wfn 5412   -1-1-onto->wf1o 5416   ` cfv 5417   Basecbs 13428   lecple 13495   HLchlt 29837   LHypclh 30470   DIsoAcdia 31515
This theorem is referenced by:  diaclN  31537  diacnvclN  31538  dia1elN  31541  diainN  31544  diaintclN  31545  diasslssN  31546  docaclN  31611  diaocN  31612  doca3N  31614  diaf1oN  31617
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-1st 6312  df-2nd 6313  df-undef 6506  df-riota 6512  df-map 6983  df-poset 14362  df-plt 14374  df-lub 14390  df-glb 14391  df-join 14392  df-meet 14393  df-p0 14427  df-p1 14428  df-lat 14434  df-clat 14496  df-oposet 29663  df-ol 29665  df-oml 29666  df-covers 29753  df-ats 29754  df-atl 29785  df-cvlat 29809  df-hlat 29838  df-llines 29984  df-lplanes 29985  df-lvols 29986  df-lines 29987  df-psubsp 29989  df-pmap 29990  df-padd 30282  df-lhyp 30474  df-laut 30475  df-ldil 30590  df-ltrn 30591  df-trl 30645  df-disoa 31516
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