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Theorem mppsval 31849
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 6375 . . . . . . . 8 (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3 mppsval.p . . . . . . . 8 𝑃 = (mPreSt‘𝑇)
42, 3syl6eqr 2817 . . . . . . 7 (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
54eleq2d 2830 . . . . . 6 (𝑡 = 𝑇 → (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
6 fveq2 6375 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
7 mppsval.c . . . . . . . . 9 𝐶 = (mCls‘𝑇)
86, 7syl6eqr 2817 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
98oveqd 6859 . . . . . . 7 (𝑡 = 𝑇 → (𝑑(mCls‘𝑡)) = (𝑑𝐶))
109eleq2d 2830 . . . . . 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 6907 . . . 4 (𝑡 = 𝑇 → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))} = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
13 df-mpps 31775 . . . 4 mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)))})
143fvexi 6389 . . . . 5 𝑃 ∈ V
153, 1, 7mppspstlem 31848 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
1614, 15ssexi 4964 . . . 4 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ∈ V
1712, 13, 16fvmpt 6471 . . 3 (𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
18 fvprc 6368 . . . 4 𝑇 ∈ V → (mPPSt‘𝑇) = ∅)
19 df-oprab 6846 . . . . 5 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
20 abn0 4119 . . . . . . 7 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ ↔ ∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))))
21 elfvex 6409 . . . . . . . . . . 11 (⟨𝑑, , 𝑎⟩ ∈ (mPreSt‘𝑇) → 𝑇 ∈ V)
2221, 3eleq2s 2862 . . . . . . . . . 10 (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑇 ∈ V)
2322ad2antrl 719 . . . . . . . . 9 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2423exlimivv 2027 . . . . . . . 8 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2524exlimivv 2027 . . . . . . 7 (∃𝑥𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑇 ∈ V)
2620, 25sylbi 208 . . . . . 6 ({𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ≠ ∅ → 𝑇 ∈ V)
2726necon1bi 2965 . . . . 5 𝑇 ∈ V → {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} = ∅)
2819, 27syl5eq 2811 . . . 4 𝑇 ∈ V → {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = ∅)
2918, 28eqtr4d 2802 . . 3 𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))})
3017, 29pm2.61i 176 . 2 (mPPSt‘𝑇) = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
311, 30eqtri 2787 1 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  Vcvv 3350  c0 4079  cop 4340  cotp 4342  cfv 6068  (class class class)co 6842  {coprab 6843  mPreStcmpst 31750  mClscmcls 31754  mPPStcmpps 31755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-ot 4343  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpps 31775
This theorem is referenced by:  elmpps  31850  mppspst  31851
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