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Theorem diaelrnN 31780
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 2435 . . . . 5  |-  ( Base `  K )  =  (
Base `  K )
2 eqid 2435 . . . . 5  |-  ( le
`  K )  =  ( le `  K
)
3 diaelrn.h . . . . 5  |-  H  =  ( LHyp `  K
)
4 diaelrn.i . . . . 5  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diafn 31769 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  I  Fn  { y  e.  ( Base `  K
)  |  y ( le `  K ) W } )
6 fvelrnb 5766 . . . 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 16 . . 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 4207 . . . . . 6  |-  ( y  =  x  ->  (
y ( le `  K ) W  <->  x ( le `  K ) W ) )
98elrab 3084 . . . . 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 31777 . . . . . . 7  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  ( x  e.  ( Base `  K
)  /\  x ( le `  K ) W ) )  ->  (
I `  x )  C_  T )
1211ex 424 . . . . . 6  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( ( x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( I `  x )  C_  T
) )
13 sseq1 3361 . . . . . . 7  |-  ( ( I `  x )  =  S  ->  (
( I `  x
)  C_  T  <->  S  C_  T
) )
1413biimpcd 216 . . . . . 6  |-  ( ( I `  x ) 
C_  T  ->  (
( I `  x
)  =  S  ->  S  C_  T ) )
1512, 14syl6 31 . . . . 5  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( ( x  e.  ( Base `  K
)  /\  x ( le `  K ) W )  ->  ( (
I `  x )  =  S  ->  S  C_  T ) ) )
169, 15syl5bi 209 . . . 4  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( x  e.  {
y  e.  ( Base `  K )  |  y ( le `  K
) W }  ->  ( ( I `  x
)  =  S  ->  S  C_  T ) ) )
1716rexlimdv 2821 . . 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 207 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( S  e.  ran  I  ->  S  C_  T
) )
1918imp 419 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 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701    C_ wss 3312   class class class wbr 4204   ran crn 4871    Fn wfn 5441   ` cfv 5446   Basecbs 13461   lecple 13528   LHypclh 30718   LTrncltrn 30835   DIsoAcdia 31763
This theorem is referenced by:  dvadiaN  31863  djaclN  31871  djajN  31872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-disoa 31764
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