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Theorem diaelval 38049
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 38048 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
87eleq2d 2895 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ 𝐹 ∈ {𝑓𝑇 ∣ (𝑅𝑓) 𝑋}))
9 fveq2 6663 . . . 4 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
109breq1d 5067 . . 3 (𝑓 = 𝐹 → ((𝑅𝑓) 𝑋 ↔ (𝑅𝐹) 𝑋))
1110elrab 3677 . 2 (𝐹 ∈ {𝑓𝑇 ∣ (𝑅𝑓) 𝑋} ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋))
128, 11syl6bb 288 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139   class class class wbr 5057  cfv 6348  Basecbs 16471  lecple 16560  LHypclh 37000  LTrncltrn 37117  trLctrl 37174  DIsoAcdia 38044
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-pr 5320
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-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
This theorem is referenced by:  dian0  38055  diatrl  38060  dialss  38062  diaglbN  38071  dibelval3  38163  dibopelval3  38164  diblss  38186
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