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Theorem diaelval 39056
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 39055 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
87eleq2d 2826 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ 𝐹 ∈ {𝑓𝑇 ∣ (𝑅𝑓) 𝑋}))
9 fveq2 6771 . . . 4 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
109breq1d 5089 . . 3 (𝑓 = 𝐹 → ((𝑅𝑓) 𝑋 ↔ (𝑅𝐹) 𝑋))
1110elrab 3626 . 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 205  wa 396   = wceq 1542  wcel 2110  {crab 3070   class class class wbr 5079  cfv 6432  Basecbs 16923  lecple 16980  LHypclh 38007  LTrncltrn 38124  trLctrl 38181  DIsoAcdia 39051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-disoa 39052
This theorem is referenced by:  dian0  39062  diatrl  39067  dialss  39069  diaglbN  39078  dibelval3  39170  dibopelval3  39171  diblss  39193
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