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Theorem dibn0 35117
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 2452 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2452 . . 3  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
6 eqid 2452 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibn0.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 35108 . 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 35003 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( DIsoA `  K
) `  W ) `  X )  =/=  (/) )
10 fvex 5804 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
1110mptex 6052 . . . . 5  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  e.  _V
1211snnz 4096 . . . 4  |-  { ( f  e.  ( (
LTrn `  K ) `  W )  |->  (  _I  |`  B ) ) }  =/=  (/)
139, 12jctir 538 . . 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 5360 . . 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 196 . 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 2742 1  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
I `  X )  =/=  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2645   (/)c0 3740   {csn 3980   class class class wbr 4395    |-> cmpt 4453    _I cid 4734    X. cxp 4941    |` cres 4945   ` cfv 5521   Basecbs 14287   lecple 14359   HLchlt 33314   LHypclh 33947   LTrncltrn 34064   DIsoAcdia 34992   DIsoBcdib 35102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-reu 2803  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-map 7321  df-poset 15230  df-plt 15242  df-lub 15258  df-glb 15259  df-join 15260  df-meet 15261  df-p0 15323  df-p1 15324  df-lat 15330  df-clat 15392  df-oposet 33140  df-ol 33142  df-oml 33143  df-covers 33230  df-ats 33231  df-atl 33262  df-cvlat 33286  df-hlat 33315  df-lhyp 33951  df-laut 33952  df-ldil 34067  df-ltrn 34068  df-trl 34122  df-disoa 34993  df-dib 35103
This theorem is referenced by:  dibord  35123  diblss  35134
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