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Theorem diaelval 37054
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 37053 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
87eleq2d 2864 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ 𝐹 ∈ {𝑓𝑇 ∣ (𝑅𝑓) 𝑋}))
9 fveq2 6411 . . . 4 (𝑓 = 𝐹 → (𝑅𝑓) = (𝑅𝐹))
109breq1d 4853 . . 3 (𝑓 = 𝐹 → ((𝑅𝑓) 𝑋 ↔ (𝑅𝐹) 𝑋))
1110elrab 3556 . 2 (𝐹 ∈ {𝑓𝑇 ∣ (𝑅𝑓) 𝑋} ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋))
128, 11syl6bb 279 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  {crab 3093   class class class wbr 4843  cfv 6101  Basecbs 16184  lecple 16274  LHypclh 36005  LTrncltrn 36122  trLctrl 36179  DIsoAcdia 37049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-disoa 37050
This theorem is referenced by:  dian0  37060  diatrl  37065  dialss  37067  diaglbN  37076  dibelval3  37168  dibopelval3  37169  diblss  37191
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