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Theorem dibeldmN 38453
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 2801 . . . 4 ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊)
3 dibfn.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
41, 2, 3dibdiadm 38450 . . 3 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = dom ((DIsoA‘𝐾)‘𝑊))
54eleq2d 2878 . 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 38331 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom ((DIsoA‘𝐾)‘𝑊) ↔ (𝑋𝐵𝑋 𝑊)))
95, 8bitrd 282 1 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112   class class class wbr 5033  dom cdm 5523  cfv 6328  Basecbs 16479  lecple 16568  LHypclh 37279  DIsoAcdia 38323  DIsoBcdib 38433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-disoa 38324  df-dib 38434
This theorem is referenced by:  dibf11N  38456  dibintclN  38462  dihmeetlem2N  38594
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