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Theorem mppspst 35558
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 2734 . . 3 (mCls‘𝑇) = (mCls‘𝑇)
41, 2, 3mppsval 35556 . 2 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑(mCls‘𝑇)))}
51, 2, 3mppspstlem 35555 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑(mCls‘𝑇)))} ⊆ 𝑃
64, 5eqsstri 4029 1 𝐽𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1536  wcel 2105  wss 3962  cotp 4638  cfv 6562  (class class class)co 7430  {coprab 7431  mPreStcmpst 35457  mClscmcls 35461  mPPStcmpps 35462
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpps 35482
This theorem is referenced by:  elmthm  35560  mthmpps  35566  mclspps  35568
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