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Theorem diaelval 41035
Description: Member of the partial isomorphism A for a lattice 𝐾. (Contributed by NM, 3-Dec-2013.)
Hypotheses
Ref Expression
diaval.b 𝐵 = (Base‘𝐾)
diaval.l = (le‘𝐾)
diaval.h 𝐻 = (LHyp‘𝐾)
diaval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
diaval.r 𝑅 = ((trL‘𝐾)‘𝑊)
diaval.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diaelval (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋)))

Proof of Theorem diaelval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 diaval.b . . . 4 𝐵 = (Base‘𝐾)
2 diaval.l . . . 4 = (le‘𝐾)
3 diaval.h . . . 4 𝐻 = (LHyp‘𝐾)
4 diaval.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
5 diaval.r . . . 4 𝑅 = ((trL‘𝐾)‘𝑊)
6 diaval.i . . . 4 𝐼 = ((DIsoA‘𝐾)‘𝑊)
71, 2, 3, 4, 5, 6diaval 41034 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
87eleq2d 2827 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ 𝐹 ∈ {𝑓𝑇 ∣ (𝑅𝑓) 𝑋}))
9 fveq2 6906 . . . 4 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
109breq1d 5153 . . 3 (𝑓 = 𝐹 → ((𝑅𝑓) 𝑋 ↔ (𝑅𝐹) 𝑋))
1110elrab 3692 . 2 (𝐹 ∈ {𝑓𝑇 ∣ (𝑅𝑓) 𝑋} ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋))
128, 11bitrdi 287 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {crab 3436   class class class wbr 5143  cfv 6561  Basecbs 17247  lecple 17304  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  DIsoAcdia 41030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-disoa 41031
This theorem is referenced by:  dian0  41041  diatrl  41046  dialss  41048  diaglbN  41057  dibelval3  41149  dibopelval3  41150  diblss  41172
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