Theorem mclsrcl 36143

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 4286	. . 3 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2		mclsval.c	. . . . . 6 ⊢ 𝐶 = (mCls‘𝑇)
3		fvprc 6871	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mCls‘𝑇) = ∅)
4	2, 3	eqtrid 2807	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐶 = ∅)
5	4	oveqd 7431	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾∅𝐵))
6		0ov 7451	. . . 4 ⊢ (𝐾∅𝐵) = ∅
7	5, 6	eqtrdi 2811	. . 3 ⊢ (¬ 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
8	1, 7	nsyl2 142	. 2 ⊢ (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9		fveq2 6879	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
10	9, 2	eqtr4di 2813	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
11	10	oveqd 7431	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
12	11	eleq2d 2846	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13		fvex 6892	. . . . . . . . 9 ⊢ (mDV‘𝑡) ∈ V
14	13	elpw2 5299	. . . . . . . 8 ⊢ (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16		mclsval.d	. . . . . . . . . 10 ⊢ 𝐷 = (mDV‘𝑇)
17	15, 16	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
18	17	sseq2d 3963	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾 ⊆ 𝐷))
19	14, 18	bitrid 286	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ 𝐷))
20		fvex 6892	. . . . . . . . 9 ⊢ (mEx‘𝑡) ∈ V
21	20	elpw2 5299	. . . . . . . 8 ⊢ (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22		fveq2 6879	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23		mclsval.e	. . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇)
24	22, 23	eqtr4di 2813	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
25	24	sseq2d 3963	. . . . . . . 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 3454	. . . . . . 7 ⊢ 𝑡 ∈ V
30	13	pwex 5345	. . . . . . . 8 ⊢ 𝒫 (mDV‘𝑡) ∈ V
31	20	pwex 5345	. . . . . . . 8 ⊢ 𝒫 (mEx‘𝑡) ∈ V
32	30, 31	mpoex 8079	. . . . . . 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 36079	. . . . . . . 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 6999	. . . . . . 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 7656	. . . . 5 ⊢ (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
37	28, 36	vtoclg 3517	. . . 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 1568   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  Vcvv 3450   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  ⟨cotp 4592  ∩ cint 4907   class class class wbr 5103   × cxp 5653  ran crn 5656   “ cima 5658  ‘cfv 6533  (class class class)co 7414   ∈ cmpo 7416  mAxcmax 36047  mExcmex 36049  mDVcmdv 36050  mVarscmvrs 36051  mSubstcmsub 36053  mVHcmvh 36054  mClscmcls 36059

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-mcls 36079

This theorem is used by: (None)