Theorem mppsval 33434

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 6756	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3		mppsval.p	. . . . . . . 8 ⊢ 𝑃 = (mPreSt‘𝑇)
4	2, 3	eqtr4di 2797	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5	4	eleq2d 2824	. . . . . 6 ⊢ (𝑡 = 𝑇 → (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ↔ ⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃))
6		fveq2 6756	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
7		mppsval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCls‘𝑇)
8	6, 7	eqtr4di 2797	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
9	8	oveqd 7272	. . . . . . 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 630	. . . . 5 ⊢ (𝑡 = 𝑇 → ((⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ)) ↔ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))))
12	11	oprabbidv 7319	. . . 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 33360	. . . 4 ⊢ mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
14	3	fvexi 6770	. . . . 5 ⊢ 𝑃 ∈ V
15	3, 1, 7	mppspstlem 33433	. . . . 5 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃
16	14, 15	ssexi 5241	. . . 4 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ∈ V
17	12, 13, 16	fvmpt 6857	. . 3 ⊢ (𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
18		fvprc 6748	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mPPSt‘𝑇) = ∅)
19		df-oprab 7259	. . . . 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 4311	. . . . . . 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 6789	. . . . . . . . . . 11 ⊢ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑇) → 𝑇 ∈ V)
22	21, 3	eleq2s 2857	. . . . . . . . . 10 ⊢ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 → 𝑇 ∈ V)
23	22	ad2antrl 724	. . . . . . . . 9 ⊢ ((𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑇 ∈ V)
24	23	exlimivv 1936	. . . . . . . 8 ⊢ (∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑇 ∈ V)
25	24	exlimivv 1936	. . . . . . 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 2971	. . . . 5 ⊢ (¬ 𝑇 ∈ V → {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} = ∅)
28	19, 27	syl5eq 2791	. . . 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 395   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  Vcvv 3422  ∅c0 4253  ⟨cop 4564  ⟨cotp 4566  ‘cfv 6418  (class class class)co 7255  {coprab 7256  mPreStcmpst 33335  mClscmcls 33339  mPPStcmpps 33340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-ot 4567  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpps 33360

This theorem is referenced by:  elmpps  33435  mppspst  33436