Theorem mppspst 36519

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 2763	. . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
4	1, 2, 3	mppsval 36517	. 2 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))}
5	1, 2, 3	mppspstlem 36516	. 2 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃
6	4, 5	eqsstri 3959	1 ⊢ 𝐽 ⊆ 𝑃

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ⟨cotp 4861  ‘cfv 6561  (class class class)co 7446  {coprab 7447  mPreStcmpst 36418  mClscmcls 36422  mPPStcmpps 36423

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-ot 4862  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpps 36443

This theorem is used by:  elmthm  36521  mthmpps  36527  mclspps  36529