Theorem diadmclN 38167

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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ 𝐵)

Proof of Theorem diadmclN

Step	Hyp	Ref	Expression
1		diadmcl.b	. . 3 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2821	. . 3 ⊢ (le‘𝐾) = (le‘𝐾)
3		diadmcl.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		diadmcl.i	. . 3 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
5	1, 2, 3, 4	diaeldm 38166	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋(le‘𝐾)𝑊)))
6	5	simprbda 501	1 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110   class class class wbr 5059  dom cdm 5550  ‘cfv 6350  Basecbs 16477  lecple 16566  LHypclh 37114  DIsoAcdia 38158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pr 5322

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-disoa 38159

This theorem is referenced by:  diameetN  38186  docaclN  38254  diaocN  38255  doca2N  38256  djajN  38267