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Theorem mppspstlem 33049
 Description: Lemma for mppspst 33052. (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 7154 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
2 df-ot 4531 . . . . . . . . . 10 𝑑, , 𝑎⟩ = ⟨⟨𝑑, ⟩, 𝑎
32eqeq2i 2771 . . . . . . . . 9 (𝑥 = ⟨𝑑, , 𝑎⟩ ↔ 𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩)
43biimpri 231 . . . . . . . 8 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → 𝑥 = ⟨𝑑, , 𝑎⟩)
54eleq1d 2836 . . . . . . 7 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → (𝑥𝑃 ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
65biimpar 481 . . . . . 6 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ ⟨𝑑, , 𝑎⟩ ∈ 𝑃) → 𝑥𝑃)
76adantrr 716 . . . . 5 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
87exlimiv 1931 . . . 4 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
98exlimivv 1933 . . 3 (∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
109abssi 3974 . 2 {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ⊆ 𝑃
111, 10eqsstri 3926 1 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735   ⊆ wss 3858  ⟨cop 4528  ⟨cotp 4530  ‘cfv 6335  (class class class)co 7150  {coprab 7151  mPreStcmpst 32951  mClscmcls 32955  mPPStcmpps 32956 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-ot 4531  df-oprab 7154 This theorem is referenced by:  mppsval  33050  mppspst  33052
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