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Theorem diaelrnN 31857
Description: Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diaelrn.h  |-  H  =  ( LHyp `  K
)
diaelrn.t  |-  T  =  ( ( LTrn `  K
) `  W )
diaelrn.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diaelrnN  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  S  e.  ran  I )  ->  S  C_  T )

Proof of Theorem diaelrnN
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2296 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
3 diaelrn.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 diaelrn.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diafn 31846 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { y  e.  ( Base `  K
)  |  y ( le `  K ) W } )
6 fvelrnb 5586 . . . 4  |-  ( I  Fn  { y  e.  ( Base `  K
)  |  y ( le `  K ) W }  ->  ( S  e.  ran  I  <->  E. x  e.  { y  e.  (
Base `  K )  |  y ( le
`  K ) W }  ( I `  x )  =  S ) )
75, 6syl 15 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  ran  I 
<->  E. x  e.  {
y  e.  ( Base `  K )  |  y ( le `  K
) W }  (
I `  x )  =  S ) )
8 breq1 4042 . . . . . 6  |-  ( y  =  x  ->  (
y ( le `  K ) W  <->  x ( le `  K ) W ) )
98elrab 2936 . . . . 5  |-  ( x  e.  { y  e.  ( Base `  K
)  |  y ( le `  K ) W }  <->  ( x  e.  ( Base `  K
)  /\  x ( le `  K ) W ) )
10 diaelrn.t . . . . . . . 8  |-  T  =  ( ( LTrn `  K
) `  W )
111, 2, 3, 10, 4diass 31854 . . . . . . 7  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( x  e.  ( Base `  K
)  /\  x ( le `  K ) W ) )  ->  (
I `  x )  C_  T )
1211ex 423 . . . . . 6  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( ( x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( I `  x )  C_  T
) )
13 sseq1 3212 . . . . . . 7  |-  ( ( I `  x )  =  S  ->  (
( I `  x
)  C_  T  <->  S  C_  T
) )
1413biimpcd 215 . . . . . 6  |-  ( ( I `  x ) 
C_  T  ->  (
( I `  x
)  =  S  ->  S  C_  T ) )
1512, 14syl6 29 . . . . 5  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( ( x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( (
I `  x )  =  S  ->  S  C_  T ) ) )
169, 15syl5bi 208 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( x  e.  {
y  e.  ( Base `  K )  |  y ( le `  K
) W }  ->  ( ( I `  x
)  =  S  ->  S  C_  T ) ) )
1716rexlimdv 2679 . . 3  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( E. x  e. 
{ y  e.  (
Base `  K )  |  y ( le
`  K ) W }  ( I `  x )  =  S  ->  S  C_  T
) )
187, 17sylbid 206 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  ran  I  ->  S  C_  T
) )
1918imp 418 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  S  e.  ran  I )  ->  S  C_  T )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557   {crab 2560    C_ wss 3165   class class class wbr 4039   ran crn 4706    Fn wfn 5266   ` cfv 5271   Basecbs 13164   lecple 13231   LHypclh 30795   LTrncltrn 30912   DIsoAcdia 31840
This theorem is referenced by:  dvadiaN  31940  djaclN  31948  djajN  31949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-disoa 31841
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