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Theorem dicelval3 38315
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 38314 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)})
109eleq2d 2898 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)}))
11 excom 2165 . . . 4 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
12 an12 643 . . . . . . 7 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
1312exbii 1844 . . . . . 6 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
14 fvex 6682 . . . . . . 7 (𝑠𝐺) ∈ V
15 opeq1 4802 . . . . . . . . 9 (𝑓 = (𝑠𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠𝐺), 𝑠⟩)
1615eqeq2d 2832 . . . . . . . 8 (𝑓 = (𝑠𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
1716anbi1d 631 . . . . . . 7 (𝑓 = (𝑠𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸)))
1814, 17ceqsexv 3541 . . . . . 6 (∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸))
19 ancom 463 . . . . . 6 ((𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2013, 18, 193bitri 299 . . . . 5 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2120exbii 1844 . . . 4 (∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2211, 21bitri 277 . . 3 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
23 elopab 5413 . . 3 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
24 df-rex 3144 . . 3 (∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2522, 23, 243bitr4i 305 . 2 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩)
2610, 25syl6bb 289 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398   = wceq 1533  wex 1776  wcel 2110  wrex 3139  cop 4572   class class class wbr 5065  {copab 5127  cfv 6354  crio 7112  lecple 16571  occoc 16572  Atomscatm 36398  LHypclh 37119  LTrncltrn 37236  TEndoctendo 37887  DIsoCcdic 38307
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-dic 38308
This theorem is referenced by:  cdlemn11pre  38345  dihord2pre  38360
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