Theorem mclsrcl 36450

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 4289	. . 3 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2		mclsval.c	. . . . . 6 ⊢ 𝐶 = (mCls‘𝑇)
3		fvprc 6888	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mCls‘𝑇) = ∅)
4	2, 3	eqtrid 2810	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐶 = ∅)
5	4	oveqd 7447	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾∅𝐵))
6		0ov 7467	. . . 4 ⊢ (𝐾∅𝐵) = ∅
7	5, 6	eqtrdi 2814	. . 3 ⊢ (¬ 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
8	1, 7	nsyl2 142	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9		fveq2 6896	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
10	9, 2	eqtr4di 2816	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
11	10	oveqd 7447	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
12	11	eleq2d 2850	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13		fvex 6909	. . . . . . . . 9 ⊢ (mDV‘𝑡) ∈ V
14	13	elpw2 5300	. . . . . . . 8 ⊢ (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16		mclsval.d	. . . . . . . . . 10 ⊢ 𝐷 = (mDV‘𝑇)
17	15, 16	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
18	17	sseq2d 3945	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾 ⊆ 𝐷))
19	14, 18	bitrid 286	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ 𝐷))
20		fvex 6909	. . . . . . . . 9 ⊢ (mEx‘𝑡) ∈ V
21	20	elpw2 5300	. . . . . . . 8 ⊢ (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22		fveq2 6896	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23		mclsval.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
24	22, 23	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
25	24	sseq2d 3945	. . . . . . . 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 644	. . . . . 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 3458	. . . . . . 7 ⊢ 𝑡 ∈ V
30	13	pwex 5346	. . . . . . . 8 ⊢ 𝒫 (mDV‘𝑡) ∈ V
31	20	pwex 5346	. . . . . . . 8 ⊢ 𝒫 (mEx‘𝑡) ∈ V
32	30, 31	mpoex 8103	. . . . . . 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 36386	. . . . . . . 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 7016	. . . . . . 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 705	. . . . . 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 7672	. . . . 5 ⊢ (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
37	28, 36	vtoclg 3521	. . . 4 ⊢ (𝑇 ∈ V → (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸)))
38	8, 37	mpcom 39	. . 3 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))
39	38	simpld 500	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝐾 ⊆ 𝐷)
40	38	simprd 501	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝐵 ⊆ 𝐸)
41	8, 39, 40	3jca 1146	1 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾 ⊆ 𝐷 ∧ 𝐵 ⊆ 𝐸))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569   ∈ wcel 2146  {cab 2741  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  𝒫 cpw 4560  ⟨cotp 4861  ∩ cint 4910   class class class wbr 5106   × cxp 5653  ran crn 5656   “ cima 5658  ‘cfv 6547  (class class class)co 7430   ∈ cmpo 7432  mAxcmax 36354  mExcmex 36356  mDVcmdv 36357  mVarscmvrs 36358  mSubstcmsub 36360  mVHcmvh 36361  mClscmcls 36366

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8011  df-2nd 8012  df-mcls 36386

This theorem is used by: (None)