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Theorem dibn0 31268
Description: The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.)
Hypotheses
Ref Expression
dibn0.b  |-  B  =  ( Base `  K
)
dibn0.l  |-  .<_  =  ( le `  K )
dibn0.h  |-  H  =  ( LHyp `  K
)
dibn0.i  |-  I  =  ( ( DIsoB `  K
) `  W )
Assertion
Ref Expression
dibn0  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )

Proof of Theorem dibn0
Dummy variable  f is distinct from all other variables.
StepHypRef Expression
1 dibn0.b . . 3  |-  B  =  ( Base `  K
)
2 dibn0.l . . 3  |-  .<_  =  ( le `  K )
3 dibn0.h . . 3  |-  H  =  ( LHyp `  K
)
4 eqid 2387 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2387 . . 3  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
6 eqid 2387 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibn0.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 31259 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =  ( ( ( ( DIsoA `  K ) `  W ) `  X
)  X.  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) } ) )
91, 2, 3, 6dian0 31154 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( DIsoA `  K
) `  W ) `  X )  =/=  (/) )
10 fvex 5682 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
1110mptex 5905 . . . . 5  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  e.  _V
1211snnz 3865 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) }  =/=  (/)
139, 12jctir 525 . . 3  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( ( DIsoA `  K ) `  W
) `  X )  =/=  (/)  /\  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) }  =/=  (/) ) )
14 xpnz 5232 . . 3  |-  ( ( ( ( ( DIsoA `  K ) `  W
) `  X )  =/=  (/)  /\  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) }  =/=  (/) )  <->  ( (
( ( DIsoA `  K
) `  W ) `  X )  X.  {
( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  =/=  (/) )
1513, 14sylib 189 . 2  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( ( DIsoA `  K ) `  W
) `  X )  X.  { ( f  e.  ( ( LTrn `  K
) `  W )  |->  (  _I  |`  B ) ) } )  =/=  (/) )
168, 15eqnetrd 2568 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   (/)c0 3571   {csn 3757   class class class wbr 4153    e. cmpt 4207    _I cid 4434    X. cxp 4816    |` cres 4820   ` cfv 5394   Basecbs 13396   lecple 13463   HLchlt 29465   LHypclh 30098   LTrncltrn 30215   DIsoAcdia 31143   DIsoBcdib 31253
This theorem is referenced by:  dibord  31274  diblss  31285
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-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-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-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-lhyp 30102  df-laut 30103  df-ldil 30218  df-ltrn 30219  df-trl 30273  df-disoa 31144  df-dib 31254
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