Theorem mppspst 36308

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 2761	. . 3 ⊢ (mCls‘𝑇) = (mCls‘𝑇)
4	1, 2, 3	mppsval 36306	. 2 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))}
5	1, 2, 3	mppspstlem 36305	. 2 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑(mCls‘𝑇)ℎ))} ⊆ 𝑃
6	4, 5	eqsstri 3977	1 ⊢ 𝐽 ⊆ 𝑃

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ⟨cotp 4592  ‘cfv 6531  (class class class)co 7412  {coprab 7413  mPreStcmpst 36207  mClscmcls 36211  mPPStcmpps 36212

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-ot 4593  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpps 36232

This theorem is used by:  elmthm  36310  mthmpps  36316  mclspps  36318