Theorem mppspst 31988

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 2799	. . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
4	1, 2, 3	mppsval 31986	. 2 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))}
5	1, 2, 3	mppspstlem 31985	. 2 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃
6	4, 5	eqsstri 3831	1 ⊢ 𝐽 ⊆ 𝑃

Syntax hints:   ∧ wa 385   = wceq 1653   ∈ wcel 2157   ⊆ wss 3769  ⟨cotp 4376  ‘cfv 6101  (class class class)co 6878  {coprab 6879  mPreStcmpst 31887  mClscmcls 31891  mPPStcmpps 31892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-ot 4377  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpps 31912

This theorem is referenced by:  elmthm  31990  mthmpps  31996  mclspps  31998