Theorem mppspst 33436

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	⊢ 𝐽 ⊆ 𝑃

Proof of Theorem mppspst

Dummy variables 𝑎 𝑑 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mppsval.p	. . 3 ⊢ 𝑃 = (mPreSt‘𝑇)
2		mppsval.j	. . 3 ⊢ 𝐽 = (mPPSt‘𝑇)
3		eqid 2738	. . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
4	1, 2, 3	mppsval 33434	. 2 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))}
5	1, 2, 3	mppspstlem 33433	. 2 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃
6	4, 5	eqsstri 3951	1 ⊢ 𝐽 ⊆ 𝑃

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2108   ⊆ wss 3883  ⟨cotp 4566  ‘cfv 6418  (class class class)co 7255  {coprab 7256  mPreStcmpst 33335  mClscmcls 33339  mPPStcmpps 33340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-ot 4567  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpps 33360

This theorem is referenced by:  elmthm  33438  mthmpps  33444  mclspps  33446