Theorem mppspstlem 35784

Description: Lemma for mppspst 35787. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mppsval.p	⊢ 𝑃 = (mPreSt‘𝑇)
mppsval.j	⊢ 𝐽 = (mPPSt‘𝑇)
mppsval.c	⊢ 𝐶 = (mCls‘𝑇)

Assertion

Ref	Expression
mppspstlem	⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃

Distinct variable groups:   𝑎,𝑑,ℎ,𝐶   𝑃,𝑎,𝑑,ℎ   𝑇,𝑎,𝑑,ℎ

Allowed substitution hints:   𝐽(ℎ,𝑎,𝑑)

Proof of Theorem mppspstlem

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-oprab 7372	. 2 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} = {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))}
2		df-ot 4591	. . . . . . . . . 10 ⊢ ⟨𝑑, ℎ, 𝑎⟩ = ⟨⟨𝑑, ℎ⟩, 𝑎⟩
3	2	eqeq2i 2750	. . . . . . . . 9 ⊢ (𝑥 = ⟨𝑑, ℎ, 𝑎⟩ ↔ 𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩)
4	3	biimpri 228	. . . . . . . 8 ⊢ (𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ → 𝑥 = ⟨𝑑, ℎ, 𝑎⟩)
5	4	eleq1d 2822	. . . . . . 7 ⊢ (𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ → (𝑥 ∈ 𝑃 ↔ ⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃))
6	5	biimpar 477	. . . . . 6 ⊢ ((𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ ⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃) → 𝑥 ∈ 𝑃)
7	6	adantrr 718	. . . . 5 ⊢ ((𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃)
8	7	exlimiv 1932	. . . 4 ⊢ (∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃)
9	8	exlimivv 1934	. . 3 ⊢ (∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃)
10	9	abssi 4022	. 2 ⊢ {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} ⊆ 𝑃
11	1, 10	eqsstri 3982	1 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ⊆ wss 3903  ⟨cop 4588  ⟨cotp 4590  ‘cfv 6500  (class class class)co 7368  {coprab 7369  mPreStcmpst 35686  mClscmcls 35690  mPPStcmpps 35691

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ss 3920  df-ot 4591  df-oprab 7372

This theorem is referenced by:  mppsval  35785  mppspst  35787