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Theorem dicelval3 38840
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 38839 . . 3 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)})
109eleq2d 2819 . 2 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ 𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)}))
11 excom 2170 . . . 4 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
12 an12 645 . . . . . . 7 ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
1312exbii 1854 . . . . . 6 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)))
14 fvex 6690 . . . . . . 7 (𝑠𝐺) ∈ V
15 opeq1 4760 . . . . . . . . 9 (𝑓 = (𝑠𝐺) → ⟨𝑓, 𝑠⟩ = ⟨(𝑠𝐺), 𝑠⟩)
1615eqeq2d 2750 . . . . . . . 8 (𝑓 = (𝑠𝐺) → (𝑌 = ⟨𝑓, 𝑠⟩ ↔ 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
1716anbi1d 633 . . . . . . 7 (𝑓 = (𝑠𝐺) → ((𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸)))
1814, 17ceqsexv 3446 . . . . . 6 (∃𝑓(𝑓 = (𝑠𝐺) ∧ (𝑌 = ⟨𝑓, 𝑠⟩ ∧ 𝑠𝐸)) ↔ (𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸))
19 ancom 464 . . . . . 6 ((𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ∧ 𝑠𝐸) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2013, 18, 193bitri 300 . . . . 5 (∃𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ (𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2120exbii 1854 . . . 4 (∃𝑠𝑓(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2211, 21bitri 278 . . 3 (∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)) ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
23 elopab 5383 . . 3 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑓𝑠(𝑌 = ⟨𝑓, 𝑠⟩ ∧ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)))
24 df-rex 3060 . . 3 (∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩ ↔ ∃𝑠(𝑠𝐸𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
2522, 23, 243bitr4i 306 . 2 (𝑌 ∈ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)} ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩)
2610, 25bitrdi 290 1 (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399   = wceq 1542  wex 1786  wcel 2114  wrex 3055  cop 4523   class class class wbr 5031  {copab 5093  cfv 6340  crio 7129  lecple 16678  occoc 16679  Atomscatm 36923  LHypclh 37644  LTrncltrn 37761  TEndoctendo 38412  DIsoCcdic 38832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297  ax-un 7482
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-reu 3061  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-pw 4491  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-ima 5539  df-iota 6298  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7130  df-dic 38833
This theorem is referenced by:  cdlemn11pre  38870  dihord2pre  38885
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