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Theorem diaeldm 41019
Description: Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)
Hypotheses
Ref Expression
diafn.b 𝐵 = (Base‘𝐾)
diafn.l = (le‘𝐾)
diafn.h 𝐻 = (LHyp‘𝐾)
diafn.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diaeldm ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋 𝑊)))

Proof of Theorem diaeldm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 diafn.b . . . 4 𝐵 = (Base‘𝐾)
2 diafn.l . . . 4 = (le‘𝐾)
3 diafn.h . . . 4 𝐻 = (LHyp‘𝐾)
4 diafn.i . . . 4 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diadm 41018 . . 3 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
65eleq2d 2825 . 2 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼𝑋 ∈ {𝑥𝐵𝑥 𝑊}))
7 breq1 5151 . . 3 (𝑥 = 𝑋 → (𝑥 𝑊𝑋 𝑊))
87elrab 3695 . 2 (𝑋 ∈ {𝑥𝐵𝑥 𝑊} ↔ (𝑋𝐵𝑋 𝑊))
96, 8bitrdi 287 1 ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋 𝑊)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  {crab 3433   class class class wbr 5148  dom cdm 5689  cfv 6563  Basecbs 17245  lecple 17305  LHypclh 39967  DIsoAcdia 41011
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-disoa 41012
This theorem is referenced by:  diadmclN  41020  diadmleN  41021  dia0eldmN  41023  dia1eldmN  41024  diaf11N  41032  diaglbN  41038  diaintclN  41041  diasslssN  41042  docaclN  41107  doca2N  41109  djajN  41120  dibval2  41127  dibeldmN  41141
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