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Theorem elmpps 31808
Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mppsval.p 𝑃 = (mPreSt‘𝑇)
mppsval.j 𝐽 = (mPPSt‘𝑇)
mppsval.c 𝐶 = (mCls‘𝑇)
Assertion
Ref Expression
elmpps (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))

Proof of Theorem elmpps
Dummy variables 𝑎 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ot 4325 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
2 mppsval.p . . . 4 𝑃 = (mPreSt‘𝑇)
3 mppsval.j . . . 4 𝐽 = (mPPSt‘𝑇)
4 mppsval.c . . . 4 𝐶 = (mCls‘𝑇)
52, 3, 4mppsval 31807 . . 3 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
61, 5eleq12i 2843 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
7 oprabss 6893 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ ((V × V) × V)
87sseli 3748 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
92mpstssv 31774 . . . . . 6 𝑃 ⊆ ((V × V) × V)
109sseli 3748 . . . . 5 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
111, 10syl5eqelr 2855 . . . 4 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
1211adantr 466 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)) → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
13 opelxp 5286 . . . 4 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
14 opelxp 5286 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
15 simp1 1130 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → 𝑑 = 𝐷)
16 simp2 1131 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → = 𝐻)
17 simp3 1132 . . . . . . . . . 10 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → 𝑎 = 𝐴)
1815, 16, 17oteq123d 4554 . . . . . . . . 9 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → ⟨𝑑, , 𝑎⟩ = ⟨𝐷, 𝐻, 𝐴⟩)
1918eleq1d 2835 . . . . . . . 8 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (⟨𝑑, , 𝑎⟩ ∈ 𝑃 ↔ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃))
2015, 16oveq12d 6811 . . . . . . . . 9 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (𝑑𝐶) = (𝐷𝐶𝐻))
2117, 20eleq12d 2844 . . . . . . . 8 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → (𝑎 ∈ (𝑑𝐶) ↔ 𝐴 ∈ (𝐷𝐶𝐻)))
2219, 21anbi12d 616 . . . . . . 7 ((𝑑 = 𝐷 = 𝐻𝑎 = 𝐴) → ((⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)) ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2322eloprabga 6894 . . . . . 6 ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
24233expa 1111 . . . . 5 (((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2514, 24sylanb 570 . . . 4 ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
2613, 25sylbi 207 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) → (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻))))
278, 12, 26pm5.21nii 367 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))
286, 27bitri 264 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝐽 ↔ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃𝐴 ∈ (𝐷𝐶𝐻)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382  w3a 1071   = wceq 1631  wcel 2145  Vcvv 3351  cop 4322  cotp 4324   × cxp 5247  cfv 6031  (class class class)co 6793  {coprab 6794  mPreStcmpst 31708  mClscmcls 31712  mPPStcmpps 31713
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-ot 4325  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpst 31728  df-mpps 31733
This theorem is referenced by:  mthmpps  31817  mclspps  31819
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