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Theorem dicelval3 39194
Description: Member of the partial isomorphism C. (Contributed by NM, 26-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)
dicval2.g 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
Assertion
Ref Expression
dicelval3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
Distinct variable groups:   𝑔,𝑠,𝐾   𝑇,𝑔   𝑔,𝑊,𝑠   𝐸,𝑠   𝑄,𝑔,𝑠   𝑌,𝑠
Allowed substitution hints:   𝐴(𝑔,𝑠)   𝑃(𝑔,𝑠)   𝑇(𝑠)   𝐸(𝑔)   𝐺(𝑔,𝑠)   𝐻(𝑔,𝑠)   𝐼(𝑔,𝑠)   (𝑔,𝑠)   𝑉(𝑔,𝑠)   𝑌(𝑔)

Proof of Theorem dicelval3
Dummy variable 𝑓 is 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)
8 dicval2.g . . . 4 𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)
91, 2, 3, 4, 5, 6, 7, 8dicval2 39193 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)})
109eleq2d 2824 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)}))
11 excom 2162 . . . 4 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
12 an12 642 . . . . . . 7 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
1312exbii 1850 . . . . . 6 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
14 fvex 6787 . . . . . . 7 (𝑠𝐺) ∈ V
15 opeq1 4804 . . . . . . . . 9 (𝑓 = (𝑠𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠𝐺), 𝑠⟩)
1615eqeq2d 2749 . . . . . . . 8 (𝑓 = (𝑠𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
1716anbi1d 630 . . . . . . 7 (𝑓 = (𝑠𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸)))
1814, 17ceqsexv 3479 . . . . . 6 (∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸))
19 ancom 461 . . . . . 6 ((𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2013, 18, 193bitri 297 . . . . 5 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2120exbii 1850 . . . 4 (∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2211, 21bitri 274 . . 3 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
23 elopab 5440 . . 3 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
24 df-rex 3070 . . 3 (∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2522, 23, 243bitr4i 303 . 2 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩)
2610, 25bitrdi 287 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1539  wex 1782  wcel 2106  wrex 3065  cop 4567   class class class wbr 5074  {copab 5136  cfv 6433  crio 7231  lecple 16969  occoc 16970  Atomscatm 37277  LHypclh 37998  LTrncltrn 38115  TEndoctendo 38766  DIsoCcdic 39186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-dic 39187
This theorem is referenced by:  cdlemn11pre  39224  dihord2pre  39239
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