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Theorem dicopelval 41179
Description: Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Feb-2014.)
Hypotheses
Ref Expression
dicval.l = (le‘𝐾)
dicval.a 𝐴 = (Atoms‘𝐾)
dicval.h 𝐻 = (LHyp‘𝐾)
dicval.p 𝑃 = ((oc‘𝐾)‘𝑊)
dicval.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dicval.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
dicval.i 𝐼 = ((DIsoC‘𝐾)‘𝑊)
dicelval.f 𝐹 ∈ V
dicelval.s 𝑆 ∈ V
Assertion
Ref Expression
dicopelval (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸)))
Distinct variable groups:   𝑔,𝐾   𝑇,𝑔   𝑔,𝑊   𝑄,𝑔
Allowed substitution hints:   𝐴(𝑔)   𝑃(𝑔)   𝑆(𝑔)   𝐸(𝑔)   𝐹(𝑔)   𝐻(𝑔)   𝐼(𝑔)   (𝑔)   𝑉(𝑔)

Proof of Theorem dicopelval
Dummy variables 𝑓 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dicval.l . . . 4 = (le‘𝐾)
2 dicval.a . . . 4 𝐴 = (Atoms‘𝐾)
3 dicval.h . . . 4 𝐻 = (LHyp‘𝐾)
4 dicval.p . . . 4 𝑃 = ((oc‘𝐾)‘𝑊)
5 dicval.t . . . 4 𝑇 = ((LTrn‘𝐾)‘𝑊)
6 dicval.e . . . 4 𝐸 = ((TEndo‘𝐾)‘𝑊)
7 dicval.i . . . 4 𝐼 = ((DIsoC‘𝐾)‘𝑊)
81, 2, 3, 4, 5, 6, 7dicval 41178 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
98eleq2d 2827 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ ⟨𝐹, 𝑆⟩ ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)}))
10 dicelval.f . . 3 𝐹 ∈ V
11 dicelval.s . . 3 𝑆 ∈ V
12 eqeq1 2741 . . . 4 (𝑓 = 𝐹 → (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ 𝐹 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
1312anbi1d 631 . . 3 (𝑓 = 𝐹 → ((𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) ↔ (𝐹 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)))
14 fveq1 6905 . . . . 5 (𝑠 = 𝑆 → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1514eqeq2d 2748 . . . 4 (𝑠 = 𝑆 → (𝐹 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ 𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
16 eleq1 2829 . . . 4 (𝑠 = 𝑆 → (𝑠𝐸𝑆𝐸))
1715, 16anbi12d 632 . . 3 (𝑠 = 𝑆 → ((𝐹 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸) ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸)))
1810, 11, 13, 17opelopab 5547 . 2 (⟨𝐹, 𝑆⟩ ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)} ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸))
199, 18bitrdi 287 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cop 4632   class class class wbr 5143  {copab 5205  cfv 6561  crio 7387  lecple 17304  occoc 17305  Atomscatm 39264  LHypclh 39986  LTrncltrn 40103  TEndoctendo 40754  DIsoCcdic 41174
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-pow 5365  ax-pr 5432  ax-un 7755
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-riota 7388  df-dic 41175
This theorem is referenced by:  dicopelval2  41183  dicvaddcl  41192  dicvscacl  41193  dicn0  41194
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