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Theorem mppspst 31778
 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 2760 . . 3 (mCls‘𝑇) = (mCls‘𝑇)
41, 2, 3mppsval 31776 . 2 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑(mCls‘𝑇)))}
51, 2, 3mppspstlem 31775 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑(mCls‘𝑇)))} ⊆ 𝑃
64, 5eqsstri 3776 1 𝐽𝑃
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1632   ∈ wcel 2139   ⊆ wss 3715  ⟨cotp 4329  ‘cfv 6049  (class class class)co 6813  {coprab 6814  mPreStcmpst 31677  mClscmcls 31681  mPPStcmpps 31682 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-iota 6012  df-fun 6051  df-fv 6057  df-ov 6816  df-oprab 6817  df-mpps 31702 This theorem is referenced by:  elmthm  31780  mthmpps  31786  mclspps  31788
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