Theorem mclsrcl 35952

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 4301	. . 3 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2		mclsval.c	. . . . . 6 ⊢ 𝐶 = (mCls‘𝑇)
3		fvprc 6874	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mCls‘𝑇) = ∅)
4	2, 3	eqtrid 2816	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐶 = ∅)
5	4	oveqd 7428	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾∅𝐵))
6		0ov 7448	. . . 4 ⊢ (𝐾∅𝐵) = ∅
7	5, 6	eqtrdi 2820	. . 3 ⊢ (¬ 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
8	1, 7	nsyl2 142	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9		fveq2 6882	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
10	9, 2	eqtr4di 2822	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
11	10	oveqd 7428	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
12	11	eleq2d 2855	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13		fvex 6895	. . . . . . . . 9 ⊢ (mDV‘𝑡) ∈ V
14	13	elpw2 5305	. . . . . . . 8 ⊢ (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16		mclsval.d	. . . . . . . . . 10 ⊢ 𝐷 = (mDV‘𝑇)
17	15, 16	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
18	17	sseq2d 3977	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾 ⊆ 𝐷))
19	14, 18	bitrid 286	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ 𝐷))
20		fvex 6895	. . . . . . . . 9 ⊢ (mEx‘𝑡) ∈ V
21	20	elpw2 5305	. . . . . . . 8 ⊢ (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22		fveq2 6882	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23		mclsval.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
24	22, 23	eqtr4di 2822	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
25	24	sseq2d 3977	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐵 ⊆ (mEx‘𝑡) ↔ 𝐵 ⊆ 𝐸))
26	21, 25	bitrid 286	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ 𝐸))
27	19, 26	anbi12d 643	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)) ↔ (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸)))
28	12, 27	imbi12d 347	. . . . 5 ⊢ (𝑡 = 𝑇 → ((𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡))) ↔ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))))
29		vex 3467	. . . . . . 7 ⊢ 𝑡 ∈ V
30	13	pwex 5352	. . . . . . . 8 ⊢ 𝒫 (mDV‘𝑡) ∈ V
31	20	pwex 5352	. . . . . . . 8 ⊢ 𝒫 (mEx‘𝑡) ∈ V
32	30, 31	mpoex 8076	. . . . . . 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 35888	. . . . . . . 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 7002	. . . . . . 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 704	. . . . . 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 7652	. . . . 5 ⊢ (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
37	28, 36	vtoclg 3531	. . . 4 ⊢ (𝑇 ∈ V → (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸)))
38	8, 37	mpcom 39	. . 3 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))
39	38	simpld 499	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝐾 ⊆ 𝐷)
40	38	simprd 500	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝐵 ⊆ 𝐸)
41	8, 39, 40	3jca 1144	1 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101  ∀wal 1565   = wceq 1567   ∈ wcel 2149  {cab 2747  ∀wral 3085  Vcvv 3463   ∪ cun 3911   ⊆ wss 3913  ∅c0 4294  𝒫 cpw 4567  ⟨cotp 4602  ∩ cint 4916   class class class wbr 5113   × cxp 5660  ran crn 5663   “ cima 5665  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413  mAxcmax 35856  mExcmex 35858  mDVcmdv 35859  mVarscmvrs 35860  mSubstcmsub 35862  mVHcmvh 35863  mClscmcls 35868

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-mcls 35888

This theorem is referenced by: (None)