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Theorem dicopelval 41160
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 41159 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
98eleq2d 2825 . 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 2739 . . . 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 6906 . . . . 5 (𝑠 = 𝑆 → (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
1514eqeq2d 2746 . . . 4 (𝑠 = 𝑆 → (𝐹 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ↔ 𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄))))
16 eleq1 2827 . . . 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 5552 . 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 1537  wcel 2106  Vcvv 3478  cop 4637   class class class wbr 5148  {copab 5210  cfv 6563  crio 7387  lecple 17305  occoc 17306  Atomscatm 39245  LHypclh 39967  LTrncltrn 40084  TEndoctendo 40735  DIsoCcdic 41155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-dic 41156
This theorem is referenced by:  dicopelval2  41164  dicvaddcl  41173  dicvscacl  41174  dicn0  41175
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