Theorem mppspst 35538

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 2734	. . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
4	1, 2, 3	mppsval 35536	. 2 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))}
5	1, 2, 3	mppspstlem 35535	. 2 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃
6	4, 5	eqsstri 4010	1 ⊢ 𝐽 ⊆ 𝑃

Syntax hints:   ∧ wa 395   = wceq 1539   ∈ wcel 2107   ⊆ wss 3931  ⟨cotp 4614  ‘cfv 6541  (class class class)co 7413  {coprab 7414  mPreStcmpst 35437  mClscmcls 35441  mPPStcmpps 35442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-ot 4615  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6494  df-fun 6543  df-fv 6549  df-ov 7416  df-oprab 7417  df-mpps 35462

This theorem is referenced by:  elmthm  35540  mthmpps  35546  mclspps  35548