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Theorem mppsval 31595
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 6229 . . . . . . . 8 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 mppsval.p . . . . . . . 8 𝑃 = (mPreSt‘𝑇)
42, 3syl6eqr 2703 . . . . . . 7 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
54eleq2d 2716 . . . . . 6 (𝑡 = 𝑇 → (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
6 fveq2 6229 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
7 mppsval.c . . . . . . . . 9 𝐶 = (mCls‘𝑇)
86, 7syl6eqr 2703 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
98oveqd 6707 . . . . . . 7 (𝑡 = 𝑇 → (𝑑(mCls‘𝑡)) = (𝑑𝐶))
109eleq2d 2716 . . . . . 6 (𝑡 = 𝑇 → (𝑎 ∈ (𝑑(mCls‘𝑡)) ↔ 𝑎 ∈ (𝑑𝐶)))
115, 10anbi12d 747 . . . . 5 (𝑡 = 𝑇 → ((⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡))) ↔ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
1211oprabbidv 6751 . . . 4 (𝑡 = 𝑇 → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))} = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
13 df-mpps 31521 . . . 4 mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))})
14 fvex 6239 . . . . . 6 (mPreSt‘𝑇) ∈ V
153, 14eqeltri 2726 . . . . 5 𝑃 ∈ V
163, 1, 7mppspstlem 31594 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
1715, 16ssexi 4836 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ∈ V
1812, 13, 17fvmpt 6321 . . 3 (𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
19 fvprc 6223 . . . 4 𝑇 ∈ V → (mPPSt‘𝑇) = ∅)
20 df-oprab 6694 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
21 abn0 3987 . . . . . . 7 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ ↔ ∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
22 elfvex 6259 . . . . . . . . . . 11 (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑇) → 𝑇 ∈ V)
2322, 3eleq2s 2748 . . . . . . . . . 10 (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑇 ∈ V)
2423ad2antrl 764 . . . . . . . . 9 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2524exlimivv 1900 . . . . . . . 8 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2625exlimivv 1900 . . . . . . 7 (∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2721, 26sylbi 207 . . . . . 6 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ → 𝑇 ∈ V)
2827necon1bi 2851 . . . . 5 𝑇 ∈ V → {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} = ∅)
2920, 28syl5eq 2697 . . . 4 𝑇 ∈ V → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = ∅)
3019, 29eqtr4d 2688 . . 3 𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
3118, 30pm2.61i 176 . 2 (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
321, 31eqtri 2673 1 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wne 2823  Vcvv 3231  c0 3948  cop 4216  cotp 4218  cfv 5926  (class class class)co 6690  {coprab 6691  mPreStcmpst 31496  mClscmcls 31500  mPPStcmpps 31501
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-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
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-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-ot 4219  df-uni 4469  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-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpps 31521
This theorem is referenced by:  elmpps  31596  mppspst  31597
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