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Theorem mppspst 32828
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 2821 . . 3 (mCls‘𝑇) = (mCls‘𝑇)
41, 2, 3mppsval 32826 . 2 𝐽 = {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑(mCls‘𝑇)))}
51, 2, 3mppspstlem 32825 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑(mCls‘𝑇)))} ⊆ 𝑃
64, 5eqsstri 3977 1 𝐽𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  wss 3910  cotp 4548  cfv 6328  (class class class)co 7130  {coprab 7131  mPreStcmpst 32727  mClscmcls 32731  mPPStcmpps 32732
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-ot 4549  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpps 32752
This theorem is referenced by:  elmthm  32830  mthmpps  32836  mclspps  32838
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