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Theorem mppsval 32893
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 6652 . . . . . . . 8 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 mppsval.p . . . . . . . 8 𝑃 = (mPreSt‘𝑇)
42, 3eqtr4di 2875 . . . . . . 7 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
54eleq2d 2899 . . . . . 6 (𝑡 = 𝑇 → (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
6 fveq2 6652 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
7 mppsval.c . . . . . . . . 9 𝐶 = (mCls‘𝑇)
86, 7eqtr4di 2875 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
98oveqd 7157 . . . . . . 7 (𝑡 = 𝑇 → (𝑑(mCls‘𝑡)) = (𝑑𝐶))
109eleq2d 2899 . . . . . 6 (𝑡 = 𝑇 → (𝑎 ∈ (𝑑(mCls‘𝑡)) ↔ 𝑎 ∈ (𝑑𝐶)))
115, 10anbi12d 633 . . . . 5 (𝑡 = 𝑇 → ((⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡))) ↔ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
1211oprabbidv 7204 . . . 4 (𝑡 = 𝑇 → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))} = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
13 df-mpps 32819 . . . 4 mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))})
143fvexi 6666 . . . . 5 𝑃 ∈ V
153, 1, 7mppspstlem 32892 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
1614, 15ssexi 5202 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ∈ V
1712, 13, 16fvmpt 6750 . . 3 (𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
18 fvprc 6645 . . . 4 𝑇 ∈ V → (mPPSt‘𝑇) = ∅)
19 df-oprab 7144 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
20 abn0 4308 . . . . . . 7 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ ↔ ∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
21 elfvex 6685 . . . . . . . . . . 11 (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑇) → 𝑇 ∈ V)
2221, 3eleq2s 2932 . . . . . . . . . 10 (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑇 ∈ V)
2322ad2antrl 727 . . . . . . . . 9 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2423exlimivv 1933 . . . . . . . 8 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2524exlimivv 1933 . . . . . . 7 (∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2620, 25sylbi 220 . . . . . 6 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ → 𝑇 ∈ V)
2726necon1bi 3039 . . . . 5 𝑇 ∈ V → {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} = ∅)
2819, 27syl5eq 2869 . . . 4 𝑇 ∈ V → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = ∅)
2918, 28eqtr4d 2860 . . 3 𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
3017, 29pm2.61i 185 . 2 (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
311, 30eqtri 2845 1 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1538  wex 1781  wcel 2114  {cab 2800  wne 3011  Vcvv 3469  c0 4265  cop 4545  cotp 4547  cfv 6334  (class class class)co 7140  {coprab 7141  mPreStcmpst 32794  mClscmcls 32798  mPPStcmpps 32799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-ot 4548  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpps 32819
This theorem is referenced by:  elmpps  32894  mppspst  32895
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