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Theorem mppspstlem 31775
Description: Lemma for mppspst 31778. (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 6817 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
2 df-ot 4330 . . . . . . . . . 10 𝑑, , 𝑎⟩ = ⟨⟨𝑑, ⟩, 𝑎
32eqeq2i 2772 . . . . . . . . 9 (𝑥 = ⟨𝑑, , 𝑎⟩ ↔ 𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩)
43biimpri 218 . . . . . . . 8 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → 𝑥 = ⟨𝑑, , 𝑎⟩)
54eleq1d 2824 . . . . . . 7 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → (𝑥𝑃 ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
65biimpar 503 . . . . . 6 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ ⟨𝑑, , 𝑎⟩ ∈ 𝑃) → 𝑥𝑃)
76adantrr 755 . . . . 5 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
87exlimiv 2007 . . . 4 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
98exlimivv 2009 . . 3 (∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
109abssi 3818 . 2 {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ⊆ 𝑃
111, 10eqsstri 3776 1 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1632  wex 1853  wcel 2139  {cab 2746  wss 3715  cop 4327  cotp 4329  cfv 6049  (class class class)co 6813  {coprab 6814  mPreStcmpst 31677  mClscmcls 31681  mPPStcmpps 31682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-in 3722  df-ss 3729  df-ot 4330  df-oprab 6817
This theorem is referenced by:  mppsval  31776  mppspst  31778
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