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Theorem diadmclN 37050
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 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋𝐵)

Proof of Theorem diadmclN
StepHypRef Expression
1 diadmcl.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2797 . . 3 (le‘𝐾) = (le‘𝐾)
3 diadmcl.h . . 3 𝐻 = (LHyp‘𝐾)
4 diadmcl.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diaeldm 37049 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋(le‘𝐾)𝑊)))
65simprbda 493 1 (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157   class class class wbr 4841  dom cdm 5310  cfv 6099  Basecbs 16181  lecple 16271  LHypclh 35997  DIsoAcdia 37041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-disoa 37042
This theorem is referenced by:  diameetN  37069  docaclN  37137  diaocN  37138  doca2N  37139  djajN  37150
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