Theorem mppspstlem 35558

Description: Lemma for mppspst 35561. (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 7393	. 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 4600	. . . . . . . . . 10 ⊢ ⟨𝑑, ℎ, 𝑎⟩ = ⟨⟨𝑑, ℎ⟩, 𝑎⟩
3	2	eqeq2i 2743	. . . . . . . . 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 2814	. . . . . . 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 4035	. 2 ⊢ {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} ⊆ 𝑃
11	1, 10	eqsstri 3995	1 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708   ⊆ wss 3916  ⟨cop 4597  ⟨cotp 4599  ‘cfv 6513  (class class class)co 7389  {coprab 7390  mPreStcmpst 35460  mClscmcls 35464  mPPStcmpps 35465

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ss 3933  df-ot 4600  df-oprab 7393

This theorem is referenced by:  mppsval  35559  mppspst  35561