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Theorem mclsrcl 31219
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 3902 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2 mclsval.c . . . . . 6 𝐶 = (mCls‘𝑇)
3 fvprc 6152 . . . . . 6 𝑇 ∈ V → (mCls‘𝑇) = ∅)
42, 3syl5eq 2667 . . . . 5 𝑇 ∈ V → 𝐶 = ∅)
54oveqd 6632 . . . 4 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾𝐵))
6 0ov 6647 . . . 4 (𝐾𝐵) = ∅
75, 6syl6eq 2671 . . 3 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
81, 7nsyl2 142 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9 fveq2 6158 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
109, 2syl6eqr 2673 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
1110oveqd 6632 . . . . . . 7 (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
1211eleq2d 2684 . . . . . 6 (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13 fvex 6168 . . . . . . . . 9 (mDV‘𝑡) ∈ V
1413elpw2 4798 . . . . . . . 8 (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15 fveq2 6158 . . . . . . . . . 10 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16 mclsval.d . . . . . . . . . 10 𝐷 = (mDV‘𝑇)
1715, 16syl6eqr 2673 . . . . . . . . 9 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
1817sseq2d 3618 . . . . . . . 8 (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾𝐷))
1914, 18syl5bb 272 . . . . . . 7 (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾𝐷))
20 fvex 6168 . . . . . . . . 9 (mEx‘𝑡) ∈ V
2120elpw2 4798 . . . . . . . 8 (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22 fveq2 6158 . . . . . . . . . 10 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23 mclsval.e . . . . . . . . . 10 𝐸 = (mEx‘𝑇)
2422, 23syl6eqr 2673 . . . . . . . . 9 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
2524sseq2d 3618 . . . . . . . 8 (𝑡 = 𝑇 → (𝐵 ⊆ (mEx‘𝑡) ↔ 𝐵𝐸))
2621, 25syl5bb 272 . . . . . . 7 (𝑡 = 𝑇 → (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵𝐸))
2719, 26anbi12d 746 . . . . . 6 (𝑡 = 𝑇 → ((𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)) ↔ (𝐾𝐷𝐵𝐸)))
2812, 27imbi12d 334 . . . . 5 (𝑡 = 𝑇 → ((𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡))) ↔ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))))
29 vex 3193 . . . . . . 7 𝑡 ∈ V
3013pwex 4818 . . . . . . . 8 𝒫 (mDV‘𝑡) ∈ V
3120pwex 4818 . . . . . . . 8 𝒫 (mEx‘𝑡) ∈ V
3230, 31mpt2ex 7207 . . . . . . 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 31155 . . . . . . . 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 6258 . . . . . . 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 707 . . . . . 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 6841 . . . . 5 (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
3728, 36vtoclg 3256 . . . 4 (𝑇 ∈ V → (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸)))
388, 37mpcom 38 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))
3938simpld 475 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐾𝐷)
4038simprd 479 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐵𝐸)
418, 39, 403jca 1240 1 (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾𝐷𝐵𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wcel 1987  {cab 2607  wral 2908  Vcvv 3190  cun 3558  wss 3560  c0 3897  𝒫 cpw 4136  cotp 4163   cint 4447   class class class wbr 4623   × cxp 5082  ran crn 5085  cima 5087  cfv 5857  (class class class)co 6615  cmpt2 6617  mAxcmax 31123  mExcmex 31125  mDVcmdv 31126  mVarscmvrs 31127  mSubstcmsub 31129  mVHcmvh 31130  mClscmcls 31135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-mcls 31155
This theorem is referenced by: (None)
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