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Theorem dibeldmN 38174
Description: Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibfn.b 𝐵 = (Base‘𝐾)
dibfn.l = (le‘𝐾)
dibfn.h 𝐻 = (LHyp‘𝐾)
dibfn.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibeldmN ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋 𝑊)))

Proof of Theorem dibeldmN
StepHypRef Expression
1 dibfn.h . . . 4 𝐻 = (LHyp‘𝐾)
2 eqid 2818 . . . 4 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
3 dibfn.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
41, 2, 3dibdiadm 38171 . . 3 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = dom ((DIsoA‘𝐾)‘𝑊))
54eleq2d 2895 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊)))
6 dibfn.b . . 3 𝐵 = (Base‘𝐾)
7 dibfn.l . . 3 = (le‘𝐾)
86, 7, 1, 2diaeldm 38052 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊) ↔ (𝑋𝐵𝑋 𝑊)))
95, 8bitrd 280 1 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105   class class class wbr 5057  dom cdm 5548  cfv 6348  Basecbs 16471  lecple 16560  LHypclh 37000  DIsoAcdia 38044  DIsoBcdib 38154
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-disoa 38045  df-dib 38155
This theorem is referenced by:  dibf11N  38177  dibintclN  38183  dihmeetlem2N  38315
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