Theorem mppsval 33534

Description: Definition of a provable pre-statement, essentially just a reorganization of the arguments of df-mcls . (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
mppsval	⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}

Distinct variable groups:   𝑎,𝑑,ℎ,𝐶   𝑃,𝑎,𝑑,ℎ   𝑇,𝑎,𝑑,ℎ

Allowed substitution hints:   𝐽(ℎ,𝑎,𝑑)

Proof of Theorem mppsval

Dummy variables 𝑡 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mppsval.j	. 2 ⊢ 𝐽 = (mPPSt‘𝑇)
2		fveq2 6774	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3		mppsval.p	. . . . . . . 8 ⊢ 𝑃 = (mPreSt‘𝑇)
4	2, 3	eqtr4di 2796	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5	4	eleq2d 2824	. . . . . 6 ⊢ (𝑡 = 𝑇 → (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ↔ ⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃))
6		fveq2 6774	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
7		mppsval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCls‘𝑇)
8	6, 7	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
9	8	oveqd 7292	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑑(mCls‘𝑡)ℎ) = (𝑑𝐶ℎ))
10	9	eleq2d 2824	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑎 ∈ (𝑑(mCls‘𝑡)ℎ) ↔ 𝑎 ∈ (𝑑𝐶ℎ)))
11	5, 10	anbi12d 631	. . . . 5 ⊢ (𝑡 = 𝑇 → ((⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ)) ↔ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))))
12	11	oprabbidv 7341	. . . 4 ⊢ (𝑡 = 𝑇 → {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))} = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
13		df-mpps 33460	. . . 4 ⊢ mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
14	3	fvexi 6788	. . . . 5 ⊢ 𝑃 ∈ V
15	3, 1, 7	mppspstlem 33533	. . . . 5 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃
16	14, 15	ssexi 5246	. . . 4 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ∈ V
17	12, 13, 16	fvmpt 6875	. . 3 ⊢ (𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
18		fvprc 6766	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mPPSt‘𝑇) = ∅)
19		df-oprab 7279	. . . . 5 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} = {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))}
20		abn0 4314	. . . . . . 7 ⊢ ({𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} ≠ ∅ ↔ ∃𝑥∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))))
21		elfvex 6807	. . . . . . . . . . 11 ⊢ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑇) → 𝑇 ∈ V)
22	21, 3	eleq2s 2857	. . . . . . . . . 10 ⊢ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 → 𝑇 ∈ V)
23	22	ad2antrl 725	. . . . . . . . 9 ⊢ ((𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑇 ∈ V)
24	23	exlimivv 1935	. . . . . . . 8 ⊢ (∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑇 ∈ V)
25	24	exlimivv 1935	. . . . . . 7 ⊢ (∃𝑥∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑇 ∈ V)
26	20, 25	sylbi 216	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} ≠ ∅ → 𝑇 ∈ V)
27	26	necon1bi 2972	. . . . 5 ⊢ (¬ 𝑇 ∈ V → {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} = ∅)
28	19, 27	eqtrid 2790	. . . 4 ⊢ (¬ 𝑇 ∈ V → {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} = ∅)
29	18, 28	eqtr4d 2781	. . 3 ⊢ (¬ 𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
30	17, 29	pm2.61i 182	. 2 ⊢ (mPPSt‘𝑇) = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}
31	1, 30	eqtri 2766	1 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}

Syntax hints:  ¬ wn 3   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  Vcvv 3432  ∅c0 4256  ⟨cop 4567  ⟨cotp 4569  ‘cfv 6433  (class class class)co 7275  {coprab 7276  mPreStcmpst 33435  mClscmcls 33439  mPPStcmpps 33440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-ot 4570  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpps 33460

This theorem is referenced by:  elmpps  33535  mppspst  33536