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Theorem mppsval 33542
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
mppsval 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
Distinct variable groups:   𝑎,𝑑,,𝐶   𝑃,𝑎,𝑑,   𝑇,𝑎,𝑑,
Allowed substitution hints:   𝐽(,𝑎,𝑑)

Proof of Theorem mppsval
Dummy variables 𝑡 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mppsval.j . 2 𝐽 = (mPPSt‘𝑇)
2 fveq2 6766 . . . . . . . 8 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 mppsval.p . . . . . . . 8 𝑃 = (mPreSt‘𝑇)
42, 3eqtr4di 2796 . . . . . . 7 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
54eleq2d 2824 . . . . . 6 (𝑡 = 𝑇 → (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
6 fveq2 6766 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
7 mppsval.c . . . . . . . . 9 𝐶 = (mCls‘𝑇)
86, 7eqtr4di 2796 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
98oveqd 7284 . . . . . . 7 (𝑡 = 𝑇 → (𝑑(mCls‘𝑡)) = (𝑑𝐶))
109eleq2d 2824 . . . . . 6 (𝑡 = 𝑇 → (𝑎 ∈ (𝑑(mCls‘𝑡)) ↔ 𝑎 ∈ (𝑑𝐶)))
115, 10anbi12d 631 . . . . 5 (𝑡 = 𝑇 → ((⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡))) ↔ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
1211oprabbidv 7331 . . . 4 (𝑡 = 𝑇 → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))} = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
13 df-mpps 33468 . . . 4 mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))})
143fvexi 6780 . . . . 5 𝑃 ∈ V
153, 1, 7mppspstlem 33541 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
1614, 15ssexi 5244 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ∈ V
1712, 13, 16fvmpt 6867 . . 3 (𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
18 fvprc 6758 . . . 4 𝑇 ∈ V → (mPPSt‘𝑇) = ∅)
19 df-oprab 7271 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
20 abn0 4314 . . . . . . 7 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ ↔ ∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
21 elfvex 6799 . . . . . . . . . . 11 (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑇) → 𝑇 ∈ V)
2221, 3eleq2s 2857 . . . . . . . . . 10 (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑇 ∈ V)
2322ad2antrl 725 . . . . . . . . 9 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2423exlimivv 1935 . . . . . . . 8 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2524exlimivv 1935 . . . . . . 7 (∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2620, 25sylbi 216 . . . . . 6 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ → 𝑇 ∈ V)
2726necon1bi 2972 . . . . 5 𝑇 ∈ V → {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} = ∅)
2819, 27eqtrid 2790 . . . 4 𝑇 ∈ V → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = ∅)
2918, 28eqtr4d 2781 . . 3 𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
3017, 29pm2.61i 182 . 2 (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
311, 30eqtri 2766 1 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wne 2943  Vcvv 3429  c0 4256  cop 4567  cotp 4569  cfv 6426  (class class class)co 7267  {coprab 7268  mPreStcmpst 33443  mClscmcls 33447  mPPStcmpps 33448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-ot 4570  df-uni 4840  df-br 5074  df-opab 5136  df-mpt 5157  df-id 5484  df-xp 5590  df-rel 5591  df-cnv 5592  df-co 5593  df-dm 5594  df-iota 6384  df-fun 6428  df-fv 6434  df-ov 7270  df-oprab 7271  df-mpps 33468
This theorem is referenced by:  elmpps  33543  mppspst  33544
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