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Theorem mclsrcl 32328
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 4179 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2 mclsval.c . . . . . 6 𝐶 = (mCls‘𝑇)
3 fvprc 6486 . . . . . 6 𝑇 ∈ V → (mCls‘𝑇) = ∅)
42, 3syl5eq 2820 . . . . 5 𝑇 ∈ V → 𝐶 = ∅)
54oveqd 6987 . . . 4 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾𝐵))
6 0ov 7006 . . . 4 (𝐾𝐵) = ∅
75, 6syl6eq 2824 . . 3 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
81, 7nsyl2 145 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9 fveq2 6493 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
109, 2syl6eqr 2826 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
1110oveqd 6987 . . . . . . 7 (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
1211eleq2d 2845 . . . . . 6 (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13 fvex 6506 . . . . . . . . 9 (mDV‘𝑡) ∈ V
1413elpw2 5098 . . . . . . . 8 (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15 fveq2 6493 . . . . . . . . . 10 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16 mclsval.d . . . . . . . . . 10 𝐷 = (mDV‘𝑇)
1715, 16syl6eqr 2826 . . . . . . . . 9 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
1817sseq2d 3883 . . . . . . . 8 (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾𝐷))
1914, 18syl5bb 275 . . . . . . 7 (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾𝐷))
20 fvex 6506 . . . . . . . . 9 (mEx‘𝑡) ∈ V
2120elpw2 5098 . . . . . . . 8 (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22 fveq2 6493 . . . . . . . . . 10 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23 mclsval.e . . . . . . . . . 10 𝐸 = (mEx‘𝑇)
2422, 23syl6eqr 2826 . . . . . . . . 9 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
2524sseq2d 3883 . . . . . . . 8 (𝑡 = 𝑇 → (𝐵 ⊆ (mEx‘𝑡) ↔ 𝐵𝐸))
2621, 25syl5bb 275 . . . . . . 7 (𝑡 = 𝑇 → (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵𝐸))
2719, 26anbi12d 621 . . . . . 6 (𝑡 = 𝑇 → ((𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)) ↔ (𝐾𝐷𝐵𝐸)))
2812, 27imbi12d 337 . . . . 5 (𝑡 = 𝑇 → ((𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡))) ↔ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))))
29 vex 3412 . . . . . . 7 𝑡 ∈ V
3013pwex 5128 . . . . . . . 8 𝒫 (mDV‘𝑡) ∈ V
3120pwex 5128 . . . . . . . 8 𝒫 (mEx‘𝑡) ∈ V
3230, 31mpoex 7579 . . . . . . 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 32264 . . . . . . . 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 6599 . . . . . . 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 679 . . . . . 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)))})
3635elmpocl 7200 . . . . 5 (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
3728, 36vtoclg 3480 . . . 4 (𝑇 ∈ V → (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸)))
388, 37mpcom 38 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))
3938simpld 487 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐾𝐷)
4038simprd 488 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐵𝐸)
418, 39, 403jca 1108 1 (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾𝐷𝐵𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 387  w3a 1068  wal 1505   = wceq 1507  wcel 2050  {cab 2752  wral 3082  Vcvv 3409  cun 3821  wss 3823  c0 4172  𝒫 cpw 4416  cotp 4443   cint 4743   class class class wbr 4923   × cxp 5399  ran crn 5402  cima 5404  cfv 6182  (class class class)co 6970  cmpo 6972  mAxcmax 32232  mExcmex 32234  mDVcmdv 32235  mVarscmvrs 32236  mSubstcmsub 32238  mVHcmvh 32239  mClscmcls 32244
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-reu 3089  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5306  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7495  df-2nd 7496  df-mcls 32264
This theorem is referenced by: (None)
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