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Theorem mppspstlem 35412
Description: Lemma for mppspst 35415. (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 7420 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
2 df-ot 4632 . . . . . . . . . 10 𝑑, , 𝑎⟩ = ⟨⟨𝑑, ⟩, 𝑎
32eqeq2i 2739 . . . . . . . . 9 (𝑥 = ⟨𝑑, , 𝑎⟩ ↔ 𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩)
43biimpri 227 . . . . . . . 8 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → 𝑥 = ⟨𝑑, , 𝑎⟩)
54eleq1d 2811 . . . . . . 7 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → (𝑥𝑃 ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
65biimpar 476 . . . . . 6 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ ⟨𝑑, , 𝑎⟩ ∈ 𝑃) → 𝑥𝑃)
76adantrr 715 . . . . 5 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
87exlimiv 1926 . . . 4 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
98exlimivv 1928 . . 3 (∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
109abssi 4063 . 2 {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ⊆ 𝑃
111, 10eqsstri 4013 1 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wss 3946  cop 4629  cotp 4631  cfv 6546  (class class class)co 7416  {coprab 7417  mPreStcmpst 35314  mClscmcls 35318  mPPStcmpps 35319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ss 3963  df-ot 4632  df-oprab 7420
This theorem is referenced by:  mppsval  35413  mppspst  35415
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