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Theorem diadmclN 39550
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 𝐡 = (Baseβ€˜πΎ)
diadmcl.h 𝐻 = (LHypβ€˜πΎ)
diadmcl.i 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
diadmclN (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ 𝑋 ∈ 𝐡)

Proof of Theorem diadmclN
StepHypRef Expression
1 diadmcl.b . . 3 𝐡 = (Baseβ€˜πΎ)
2 eqid 2733 . . 3 (leβ€˜πΎ) = (leβ€˜πΎ)
3 diadmcl.h . . 3 𝐻 = (LHypβ€˜πΎ)
4 diadmcl.i . . 3 𝐼 = ((DIsoAβ€˜πΎ)β€˜π‘Š)
51, 2, 3, 4diaeldm 39549 . 2 ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐡 ∧ 𝑋(leβ€˜πΎ)π‘Š)))
65simprbda 500 1 (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ 𝑋 ∈ 𝐡)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107   class class class wbr 5109  dom cdm 5637  β€˜cfv 6500  Basecbs 17091  lecple 17148  LHypclh 38497  DIsoAcdia 39541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-disoa 39542
This theorem is referenced by:  diameetN  39569  docaclN  39637  diaocN  39638  doca2N  39639  djajN  39650
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