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Theorem dibclN 40537
Description: Closure of partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibcl.h 𝐻 = (LHypβ€˜πΎ)
dibcl.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibclN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (πΌβ€˜π‘‹) ∈ ran 𝐼)

Proof of Theorem dibclN
StepHypRef Expression
1 dibcl.h . . . 4 𝐻 = (LHypβ€˜πΎ)
2 eqid 2724 . . . 4 ((DIsoAβ€˜πΎ)β€˜π‘Š) = ((DIsoAβ€˜πΎ)β€˜π‘Š)
3 dibcl.i . . . 4 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
41, 2, 3dibfna 40529 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn dom ((DIsoAβ€˜πΎ)β€˜π‘Š))
5 fnfun 6640 . . 3 (𝐼 Fn dom ((DIsoAβ€˜πΎ)β€˜π‘Š) β†’ Fun 𝐼)
64, 5syl 17 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ Fun 𝐼)
7 fvelrn 7069 . 2 ((Fun 𝐼 ∧ 𝑋 ∈ dom 𝐼) β†’ (πΌβ€˜π‘‹) ∈ ran 𝐼)
86, 7sylan 579 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) β†’ (πΌβ€˜π‘‹) ∈ ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  dom cdm 5667  ran crn 5668  Fun wfun 6528   Fn wfn 6529  β€˜cfv 6534  HLchlt 38724  LHypclh 39359  DIsoAcdia 40403  DIsoBcdib 40513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-dib 40514
This theorem is referenced by:  dibintclN  40542
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