Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dicelval3 Structured version   Visualization version   GIF version

Theorem dicelval3 35949
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 35948 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)})
109eleq2d 2684 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)}))
11 excom 2039 . . . 4 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
12 an12 837 . . . . . . 7 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
1312exbii 1771 . . . . . 6 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
14 fvex 6158 . . . . . . 7 (𝑠𝐺) ∈ V
15 opeq1 4370 . . . . . . . . 9 (𝑓 = (𝑠𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠𝐺), 𝑠⟩)
1615eqeq2d 2631 . . . . . . . 8 (𝑓 = (𝑠𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
1716anbi1d 740 . . . . . . 7 (𝑓 = (𝑠𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸)))
1814, 17ceqsexv 3228 . . . . . 6 (∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸))
19 ancom 466 . . . . . 6 ((𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2013, 18, 193bitri 286 . . . . 5 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2120exbii 1771 . . . 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 4943 . . 3 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
24 df-rex 2913 . . 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 384   = wceq 1480  wex 1701  wcel 1987  wrex 2908  cop 4154   class class class wbr 4613  {copab 4672  cfv 5847  crio 6564  lecple 15869  occoc 15870  Atomscatm 34030  LHypclh 34750  LTrncltrn 34867  TEndoctendo 35520  DIsoCcdic 35941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-dic 35942
This theorem is referenced by:  cdlemn11pre  35979  dihord2pre  35994
  Copyright terms: Public domain W3C validator