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Theorem mppspstlem 32704
 Description: Lemma for mppspst 32707. (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, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
Distinct variable groups:   𝑎,𝑑,,𝐶   𝑃,𝑎,𝑑,   𝑇,𝑎,𝑑,
Allowed substitution hints:   𝐽(,𝑎,𝑑)

Proof of Theorem mppspstlem
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-oprab 7155 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
2 df-ot 4572 . . . . . . . . . 10 𝑑, , 𝑎⟩ = ⟨⟨𝑑, ⟩, 𝑎
32eqeq2i 2838 . . . . . . . . 9 (𝑥 = ⟨𝑑, , 𝑎⟩ ↔ 𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩)
43biimpri 229 . . . . . . . 8 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → 𝑥 = ⟨𝑑, , 𝑎⟩)
54eleq1d 2901 . . . . . . 7 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → (𝑥𝑃 ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
65biimpar 478 . . . . . 6 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ ⟨𝑑, , 𝑎⟩ ∈ 𝑃) → 𝑥𝑃)
76adantrr 713 . . . . 5 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
87exlimiv 1924 . . . 4 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
98exlimivv 1926 . . 3 (∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
109abssi 4049 . 2 {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ⊆ 𝑃
111, 10eqsstri 4004 1 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 396   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2803   ⊆ wss 3939  ⟨cop 4569  ⟨cotp 4571  ‘cfv 6351  (class class class)co 7151  {coprab 7152  mPreStcmpst 32606  mClscmcls 32610  mPPStcmpps 32611 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-in 3946  df-ss 3955  df-ot 4572  df-oprab 7155 This theorem is referenced by:  mppsval  32705  mppspst  32707
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