Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mppspstlem Structured version   Visualization version   GIF version

Theorem mppspstlem 31939
Description: Lemma for mppspst 31942. (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 6850 . 2 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} = {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))}
2 df-ot 4345 . . . . . . . . . 10 𝑑, , 𝑎⟩ = ⟨⟨𝑑, ⟩, 𝑎
32eqeq2i 2777 . . . . . . . . 9 (𝑥 = ⟨𝑑, , 𝑎⟩ ↔ 𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩)
43biimpri 219 . . . . . . . 8 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → 𝑥 = ⟨𝑑, , 𝑎⟩)
54eleq1d 2829 . . . . . . 7 (𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ → (𝑥𝑃 ↔ ⟨𝑑, , 𝑎⟩ ∈ 𝑃))
65biimpar 469 . . . . . 6 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ ⟨𝑑, , 𝑎⟩ ∈ 𝑃) → 𝑥𝑃)
76adantrr 708 . . . . 5 ((𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
87exlimiv 2025 . . . 4 (∃𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
98exlimivv 2027 . . 3 (∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))) → 𝑥𝑃)
109abssi 3839 . 2 {𝑥 ∣ ∃𝑑𝑎(𝑥 = ⟨⟨𝑑, ⟩, 𝑎⟩ ∧ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶)))} ⊆ 𝑃
111, 10eqsstri 3797 1 {⟨⟨𝑑, ⟩, 𝑎⟩ ∣ (⟨𝑑, , 𝑎⟩ ∈ 𝑃𝑎 ∈ (𝑑𝐶))} ⊆ 𝑃
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wss 3734  cop 4342  cotp 4344  cfv 6070  (class class class)co 6846  {coprab 6847  mPreStcmpst 31841  mClscmcls 31845  mPPStcmpps 31846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-in 3741  df-ss 3748  df-ot 4345  df-oprab 6850
This theorem is referenced by:  mppsval  31940  mppspst  31942
  Copyright terms: Public domain W3C validator