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Theorem mppsval 32068
 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 6446 . . . . . . . 8 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 mppsval.p . . . . . . . 8 𝑃 = (mPreSt‘𝑇)
42, 3syl6eqr 2832 . . . . . . 7 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
54eleq2d 2845 . . . . . 6 (𝑡 = 𝑇 → (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
6 fveq2 6446 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
7 mppsval.c . . . . . . . . 9 𝐶 = (mCls‘𝑇)
86, 7syl6eqr 2832 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
98oveqd 6939 . . . . . . 7 (𝑡 = 𝑇 → (𝑑(mCls‘𝑡)) = (𝑑𝐶))
109eleq2d 2845 . . . . . 6 (𝑡 = 𝑇 → (𝑎 ∈ (𝑑(mCls‘𝑡)) ↔ 𝑎 ∈ (𝑑𝐶)))
115, 10anbi12d 624 . . . . 5 (𝑡 = 𝑇 → ((⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡))) ↔ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
1211oprabbidv 6986 . . . 4 (𝑡 = 𝑇 → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))} = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
13 df-mpps 31994 . . . 4 mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))})
143fvexi 6460 . . . . 5 𝑃 ∈ V
153, 1, 7mppspstlem 32067 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
1614, 15ssexi 5040 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ∈ V
1712, 13, 16fvmpt 6542 . . 3 (𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
18 fvprc 6439 . . . 4 𝑇 ∈ V → (mPPSt‘𝑇) = ∅)
19 df-oprab 6926 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
20 abn0 4185 . . . . . . 7 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ ↔ ∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
21 elfvex 6480 . . . . . . . . . . 11 (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑇) → 𝑇 ∈ V)
2221, 3eleq2s 2877 . . . . . . . . . 10 (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑇 ∈ V)
2322ad2antrl 718 . . . . . . . . 9 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2423exlimivv 1975 . . . . . . . 8 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2524exlimivv 1975 . . . . . . 7 (∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2620, 25sylbi 209 . . . . . 6 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ → 𝑇 ∈ V)
2726necon1bi 2997 . . . . 5 𝑇 ∈ V → {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} = ∅)
2819, 27syl5eq 2826 . . . 4 𝑇 ∈ V → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = ∅)
2918, 28eqtr4d 2817 . . 3 𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
3017, 29pm2.61i 177 . 2 (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
311, 30eqtri 2802 1 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 386   = wceq 1601  ∃wex 1823   ∈ wcel 2107  {cab 2763   ≠ wne 2969  Vcvv 3398  ∅c0 4141  ⟨cop 4404  ⟨cotp 4406  ‘cfv 6135  (class class class)co 6922  {coprab 6923  mPreStcmpst 31969  mClscmcls 31973  mPPStcmpps 31974 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-ot 4407  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpps 31994 This theorem is referenced by:  elmpps  32069  mppspst  32070
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