Theorem mppspstlem 35569

Description: Lemma for mppspst 35572. (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 7442	. 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 4643	. . . . . . . . . 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 2826	. . . . . . 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 717	. . . . 5 ⊢ ((𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃)
8	7	exlimiv 1930	. . . 4 ⊢ (∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃)
9	8	exlimivv 1932	. . 3 ⊢ (∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑥 ∈ 𝑃)
10	9	abssi 4083	. 2 ⊢ {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} ⊆ 𝑃
11	1, 10	eqsstri 4033	1 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃

Syntax hints:   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2108  {cab 2714   ⊆ wss 3966  ⟨cop 4640  ⟨cotp 4642  ‘cfv 6569  (class class class)co 7438  {coprab 7439  mPreStcmpst 35471  mClscmcls 35475  mPPStcmpps 35476

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ss 3983  df-ot 4643  df-oprab 7442

This theorem is referenced by:  mppsval  35570  mppspst  35572