Theorem mppsval 36085

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 6888	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = (mPreSt‘𝑇))
3		mppsval.p	. . . . . . . 8 ⊢ 𝑃 = (mPreSt‘𝑇)
4	2, 3	eqtr4di 2819	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mPreSt‘𝑡) = 𝑃)
5	4	eleq2d 2852	. . . . . 6 ⊢ (𝑡 = 𝑇 → (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ↔ ⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃))
6		fveq2 6888	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
7		mppsval.c	. . . . . . . . 9 ⊢ 𝐶 = (mCls‘𝑇)
8	6, 7	eqtr4di 2819	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
9	8	oveqd 7440	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝑑(mCls‘𝑡)ℎ) = (𝑑𝐶ℎ))
10	9	eleq2d 2852	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝑎 ∈ (𝑑(mCls‘𝑡)ℎ) ↔ 𝑎 ∈ (𝑑𝐶ℎ)))
11	5, 10	anbi12d 644	. . . . 5 ⊢ (𝑡 = 𝑇 → ((⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ)) ↔ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))))
12	11	oprabbidv 7489	. . . 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 36011	. . . 4 ⊢ mPPSt = (𝑡 ∈ V ↦ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑡) ∧ 𝑎 ∈ (𝑑(mCls‘𝑡)ℎ))})
14	3	fvexi 6902	. . . . 5 ⊢ 𝑃 ∈ V
15	3, 1, 7	mppspstlem 36084	. . . . 5 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ⊆ 𝑃
16	14, 15	ssexi 5298	. . . 4 ⊢ {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} ∈ V
17	12, 13, 16	fvmpt 6996	. . 3 ⊢ (𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
18		fvprc 6880	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mPPSt‘𝑇) = ∅)
19		df-oprab 7427	. . . . 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 4344	. . . . . . 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 6923	. . . . . . . . . . 11 ⊢ (⟨𝑑, ℎ, 𝑎⟩ ∈ (mPreSt‘𝑇) → 𝑇 ∈ V)
22	21, 3	eleq2s 2884	. . . . . . . . . 10 ⊢ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 → 𝑇 ∈ V)
23	22	ad2antrl 741	. . . . . . . . 9 ⊢ ((𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑇 ∈ V)
24	23	exlimivv 1965	. . . . . . . 8 ⊢ (∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑇 ∈ V)
25	24	exlimivv 1965	. . . . . . 7 ⊢ (∃𝑥∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))) → 𝑇 ∈ V)
26	20, 25	sylbi 220	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} ≠ ∅ → 𝑇 ∈ V)
27	26	necon1bi 2989	. . . . 5 ⊢ (¬ 𝑇 ∈ V → {𝑥 ∣ ∃𝑑∃ℎ∃𝑎(𝑥 = ⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∧ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ)))} = ∅)
28	19, 27	eqtrid 2813	. . . 4 ⊢ (¬ 𝑇 ∈ V → {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))} = ∅)
29	18, 28	eqtr4d 2804	. . 3 ⊢ (¬ 𝑇 ∈ V → (mPPSt‘𝑇) = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))})
30	17, 29	pm2.61i 184	. 2 ⊢ (mPPSt‘𝑇) = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}
31	1, 30	eqtri 2789	1 ⊢ 𝐽 = {⟨⟨𝑑, ℎ⟩, 𝑎⟩ ∣ (⟨𝑑, ℎ, 𝑎⟩ ∈ 𝑃 ∧ 𝑎 ∈ (𝑑𝐶ℎ))}

This proof depends on syntax axioms:  ¬ wn 3   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2744   ≠ wne 2961  Vcvv 3458  ∅c0 4289  ⟨cop 4600  ⟨cotp 4602  ‘cfv 6543  (class class class)co 7423  {coprab 7424  mPreStcmpst 35986  mClscmcls 35990  mPPStcmpps 35991

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-ot 4603  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpps 36011

This theorem is used by:  elmpps  36086  mppspst  36087