Theorem mclsrcl 35529

Description: Reverse closure for the closure function. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mclsval.d	⊢ 𝐷 = (mDV‘𝑇)
mclsval.e	⊢ 𝐸 = (mEx‘𝑇)
mclsval.c	⊢ 𝐶 = (mCls‘𝑇)

Assertion

Ref	Expression
mclsrcl	⊢ (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))

Proof of Theorem mclsrcl

Dummy variables ℎ 𝑑 𝑡 𝑐 𝑚 𝑜 𝑝 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0i 4363	. . 3 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2		mclsval.c	. . . . . 6 ⊢ 𝐶 = (mCls‘𝑇)
3		fvprc 6912	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mCls‘𝑇) = ∅)
4	2, 3	eqtrid 2792	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐶 = ∅)
5	4	oveqd 7465	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾∅𝐵))
6		0ov 7485	. . . 4 ⊢ (𝐾∅𝐵) = ∅
7	5, 6	eqtrdi 2796	. . 3 ⊢ (¬ 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
8	1, 7	nsyl2 141	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9		fveq2 6920	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
10	9, 2	eqtr4di 2798	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
11	10	oveqd 7465	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
12	11	eleq2d 2830	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13		fvex 6933	. . . . . . . . 9 ⊢ (mDV‘𝑡) ∈ V
14	13	elpw2 5352	. . . . . . . 8 ⊢ (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16		mclsval.d	. . . . . . . . . 10 ⊢ 𝐷 = (mDV‘𝑇)
17	15, 16	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
18	17	sseq2d 4041	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾 ⊆ 𝐷))
19	14, 18	bitrid 283	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ 𝐷))
20		fvex 6933	. . . . . . . . 9 ⊢ (mEx‘𝑡) ∈ V
21	20	elpw2 5352	. . . . . . . 8 ⊢ (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23		mclsval.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
24	22, 23	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
25	24	sseq2d 4041	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐵 ⊆ (mEx‘𝑡) ↔ 𝐵 ⊆ 𝐸))
26	21, 25	bitrid 283	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ 𝐸))
27	19, 26	anbi12d 631	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)) ↔ (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸)))
28	12, 27	imbi12d 344	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡))) ↔ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))))
29		vex 3492	. . . . . . 7 ⊢ 𝑡 ∈ V
30	13	pwex 5398	. . . . . . . 8 ⊢ 𝒫 (mDV‘𝑡) ∈ V
31	20	pwex 5398	. . . . . . . 8 ⊢ 𝒫 (mEx‘𝑡) ∈ V
32	30, 31	mpoex 8120	. . . . . . 7 ⊢ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}) ∈ V
33		df-mcls 35465	. . . . . . . 8 ⊢ mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
34	33	fvmpt2 7040	. . . . . . 7 ⊢ ((𝑡 ∈ V ∧ (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}) ∈ V) → (mCls‘𝑡) = (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))}))
35	29, 32, 34	mp2an 691	. . . . . 6 ⊢ (mCls‘𝑡) = (𝑑 ∈ 𝒫 (mDV‘𝑡), ℎ ∈ 𝒫 (mEx‘𝑡) ↦ ∩ {𝑐 ∣ ((ℎ ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚∀𝑜∀𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥∀𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠‘𝑝) ∈ 𝑐)))})
36	35	elmpocl 7691	. . . . 5 ⊢ (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
37	28, 36	vtoclg 3566	. . . 4 ⊢ (𝑇 ∈ V → (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸)))
38	8, 37	mpcom 38	. . 3 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))
39	38	simpld 494	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝐾 ⊆ 𝐷)
40	38	simprd 495	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝐵 ⊆ 𝐸)
41	8, 39, 40	3jca 1128	1 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1535   = wceq 1537   ∈ wcel 2108  {cab 2717  ∀wral 3067  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  ∅c0 4352  𝒫 cpw 4622  ⟨cotp 4656  ∩ cint 4970   class class class wbr 5166   × cxp 5698  ran crn 5701   “ cima 5703  ‘cfv 6573  (class class class)co 7448   ∈ cmpo 7450  mAxcmax 35433  mExcmex 35435  mDVcmdv 35436  mVarscmvrs 35437  mSubstcmsub 35439  mVHcmvh 35440  mClscmcls 35445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-mcls 35465

This theorem is referenced by: (None)