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Theorem mclsrcl 31786
 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 ∧ 𝐾𝐷𝐵𝐸))

Proof of Theorem mclsrcl
Dummy variables 𝑑 𝑡 𝑐 𝑚 𝑜 𝑝 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 4063 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2 mclsval.c . . . . . 6 𝐶 = (mCls‘𝑇)
3 fvprc 6347 . . . . . 6 𝑇 ∈ V → (mCls‘𝑇) = ∅)
42, 3syl5eq 2806 . . . . 5 𝑇 ∈ V → 𝐶 = ∅)
54oveqd 6831 . . . 4 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾𝐵))
6 0ov 6846 . . . 4 (𝐾𝐵) = ∅
75, 6syl6eq 2810 . . 3 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
81, 7nsyl2 142 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9 fveq2 6353 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
109, 2syl6eqr 2812 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
1110oveqd 6831 . . . . . . 7 (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
1211eleq2d 2825 . . . . . 6 (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13 fvex 6363 . . . . . . . . 9 (mDV‘𝑡) ∈ V
1413elpw2 4977 . . . . . . . 8 (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15 fveq2 6353 . . . . . . . . . 10 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16 mclsval.d . . . . . . . . . 10 𝐷 = (mDV‘𝑇)
1715, 16syl6eqr 2812 . . . . . . . . 9 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
1817sseq2d 3774 . . . . . . . 8 (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾𝐷))
1914, 18syl5bb 272 . . . . . . 7 (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾𝐷))
20 fvex 6363 . . . . . . . . 9 (mEx‘𝑡) ∈ V
2120elpw2 4977 . . . . . . . 8 (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22 fveq2 6353 . . . . . . . . . 10 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23 mclsval.e . . . . . . . . . 10 𝐸 = (mEx‘𝑇)
2422, 23syl6eqr 2812 . . . . . . . . 9 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
2524sseq2d 3774 . . . . . . . 8 (𝑡 = 𝑇 → (𝐵 ⊆ (mEx‘𝑡) ↔ 𝐵𝐸))
2621, 25syl5bb 272 . . . . . . 7 (𝑡 = 𝑇 → (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵𝐸))
2719, 26anbi12d 749 . . . . . 6 (𝑡 = 𝑇 → ((𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)) ↔ (𝐾𝐷𝐵𝐸)))
2812, 27imbi12d 333 . . . . 5 (𝑡 = 𝑇 → ((𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡))) ↔ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))))
29 vex 3343 . . . . . . 7 𝑡 ∈ V
3013pwex 4997 . . . . . . . 8 𝒫 (mDV‘𝑡) ∈ V
3120pwex 4997 . . . . . . . 8 𝒫 (mEx‘𝑡) ∈ V
3230, 31mpt2ex 7416 . . . . . . 7 (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}) ∈ V
33 df-mcls 31722 . . . . . . . 8 mCls = (𝑡 ∈ V ↦ (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))
3433fvmpt2 6454 . . . . . . 7 ((𝑡 ∈ V ∧ (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}) ∈ V) → (mCls‘𝑡) = (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))}))
3529, 32, 34mp2an 710 . . . . . 6 (mCls‘𝑡) = (𝑑 ∈ 𝒫 (mDV‘𝑡), ∈ 𝒫 (mEx‘𝑡) ↦ {𝑐 ∣ (( ∪ ran (mVH‘𝑡)) ⊆ 𝑐 ∧ ∀𝑚𝑜𝑝(⟨𝑚, 𝑜, 𝑝⟩ ∈ (mAx‘𝑡) → ∀𝑠 ∈ ran (mSubst‘𝑡)(((𝑠 “ (𝑜 ∪ ran (mVH‘𝑡))) ⊆ 𝑐 ∧ ∀𝑥𝑦(𝑥𝑚𝑦 → (((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑥))) × ((mVars‘𝑡)‘(𝑠‘((mVH‘𝑡)‘𝑦)))) ⊆ 𝑑)) → (𝑠𝑝) ∈ 𝑐)))})
3635elmpt2cl 7042 . . . . 5 (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
3728, 36vtoclg 3406 . . . 4 (𝑇 ∈ V → (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸)))
388, 37mpcom 38 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))
3938simpld 477 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐾𝐷)
4038simprd 482 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐵𝐸)
418, 39, 403jca 1123 1 (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾𝐷𝐵𝐸))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∧ w3a 1072  ∀wal 1630   = wceq 1632   ∈ wcel 2139  {cab 2746  ∀wral 3050  Vcvv 3340   ∪ cun 3713   ⊆ wss 3715  ∅c0 4058  𝒫 cpw 4302  ⟨cotp 4329  ∩ cint 4627   class class class wbr 4804   × cxp 5264  ran crn 5267   “ cima 5269  ‘cfv 6049  (class class class)co 6814   ↦ cmpt2 6816  mAxcmax 31690  mExcmex 31692  mDVcmdv 31693  mVarscmvrs 31694  mSubstcmsub 31696  mVHcmvh 31697  mClscmcls 31702 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-1st 7334  df-2nd 7335  df-mcls 31722 This theorem is referenced by: (None)
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