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Theorem diadmclN 31045
Description: A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
diadmcl.b  |-  B  =  ( Base `  K
)
diadmcl.h  |-  H  =  ( LHyp `  K
)
diadmcl.i  |-  I  =  ( ( DIsoA `  K
) `  W )
Assertion
Ref Expression
diadmclN  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  X  e.  B )

Proof of Theorem diadmclN
StepHypRef Expression
1 diadmcl.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2316 . . 3  |-  ( le
`  K )  =  ( le `  K
)
3 diadmcl.h . . 3  |-  H  =  ( LHyp `  K
)
4 diadmcl.i . . 3  |-  I  =  ( ( DIsoA `  K
) `  W )
51, 2, 3, 4diaeldm 31044 . 2  |-  ( ( K  e.  V  /\  W  e.  H )  ->  ( X  e.  dom  I 
<->  ( X  e.  B  /\  X ( le `  K ) W ) ) )
65simprbda 606 1  |-  ( ( ( K  e.  V  /\  W  e.  H
)  /\  X  e.  dom  I )  ->  X  e.  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1633    e. wcel 1701   class class class wbr 4060   dom cdm 4726   ` cfv 5292   Basecbs 13195   lecple 13262   LHypclh 29991   DIsoAcdia 31036
This theorem is referenced by:  diameetN  31064  docaclN  31132  diaocN  31133  doca2N  31134  djajN  31145
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-disoa 31037
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