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Theorem mppspst 35579
Description: A provable pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mppsval.p 𝑃 = (mPreSt‘𝑇)
mppsval.j 𝐽 = (mPPSt‘𝑇)
Assertion
Ref Expression
mppspst 𝐽𝑃

Proof of Theorem mppspst
Dummy variables 𝑎 𝑑 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mppsval.p . . 3 𝑃 = (mPreSt‘𝑇)
2 mppsval.j . . 3 𝐽 = (mPPSt‘𝑇)
3 eqid 2737 . . 3 (mCls‘𝑇) = (mCls‘𝑇)
41, 2, 3mppsval 35577 . 2 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑(mCls‘𝑇)))}
51, 2, 3mppspstlem 35576 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑(mCls‘𝑇)))} ⊆ 𝑃
64, 5eqsstri 4030 1 𝐽𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1540  wcel 2108  wss 3951  cotp 4634  cfv 6561  (class class class)co 7431  {coprab 7432  mPreStcmpst 35478  mClscmcls 35482  mPPStcmpps 35483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpps 35503
This theorem is referenced by:  elmthm  35581  mthmpps  35587  mclspps  35589
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