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Theorem dibn0 35950
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 2467 . . 3  |-  ( (
LTrn `  K ) `  W )  =  ( ( LTrn `  K
) `  W )
5 eqid 2467 . . 3  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  =  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )
6 eqid 2467 . . 3  |-  ( (
DIsoA `  K ) `  W )  =  ( ( DIsoA `  K ) `  W )
7 dibn0.i . . 3  |-  I  =  ( ( DIsoB `  K
) `  W )
81, 2, 3, 4, 5, 6, 7dibval2 35941 . 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 35836 . . . 4  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( X  e.  B  /\  X  .<_  W ) )  ->  (
( ( DIsoA `  K
) `  W ) `  X )  =/=  (/) )
10 fvex 5874 . . . . . 6  |-  ( (
LTrn `  K ) `  W )  e.  _V
1110mptex 6129 . . . . 5  |-  ( f  e.  ( ( LTrn `  K ) `  W
)  |->  (  _I  |`  B ) )  e.  _V
1211snnz 4145 . . . 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 5424 . . 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 2760 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 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   {csn 4027   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997    |` cres 5001   ` cfv 5586   Basecbs 14486   lecple 14558   HLchlt 34147   LHypclh 34780   LTrncltrn 34897   DIsoAcdia 35825   DIsoBcdib 35935
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-map 7419  df-poset 15429  df-plt 15441  df-lub 15457  df-glb 15458  df-join 15459  df-meet 15460  df-p0 15522  df-p1 15523  df-lat 15529  df-clat 15591  df-oposet 33973  df-ol 33975  df-oml 33976  df-covers 34063  df-ats 34064  df-atl 34095  df-cvlat 34119  df-hlat 34148  df-lhyp 34784  df-laut 34785  df-ldil 34900  df-ltrn 34901  df-trl 34955  df-disoa 35826  df-dib 35936
This theorem is referenced by:  dibord  35956  diblss  35967
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