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Theorem dicelval3 36786
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 36785 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)})
109eleq2d 2716 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)}))
11 excom 2082 . . . 4 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
12 an12 855 . . . . . . 7 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
1312exbii 1814 . . . . . 6 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
14 fvex 6239 . . . . . . 7 (𝑠𝐺) ∈ V
15 opeq1 4433 . . . . . . . . 9 (𝑓 = (𝑠𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠𝐺), 𝑠⟩)
1615eqeq2d 2661 . . . . . . . 8 (𝑓 = (𝑠𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
1716anbi1d 741 . . . . . . 7 (𝑓 = (𝑠𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸)))
1814, 17ceqsexv 3273 . . . . . 6 (∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸))
19 ancom 465 . . . . . 6 ((𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2013, 18, 193bitri 286 . . . . 5 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2120exbii 1814 . . . 4 (∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2211, 21bitri 264 . . 3 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
23 elopab 5012 . . 3 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
24 df-rex 2947 . . 3 (∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2522, 23, 243bitr4i 292 . 2 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩)
2610, 25syl6bb 276 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  cop 4216   class class class wbr 4685  {copab 4745  cfv 5926  crio 6650  lecple 15995  occoc 15996  Atomscatm 34868  LHypclh 35588  LTrncltrn 35705  TEndoctendo 36357  DIsoCcdic 36778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-dic 36779
This theorem is referenced by:  cdlemn11pre  36816  dihord2pre  36831
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