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Theorem mclsrcl 35165
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 4329 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2 mclsval.c . . . . . 6 𝐶 = (mCls‘𝑇)
3 fvprc 6883 . . . . . 6 𝑇 ∈ V → (mCls‘𝑇) = ∅)
42, 3eqtrid 2780 . . . . 5 𝑇 ∈ V → 𝐶 = ∅)
54oveqd 7431 . . . 4 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾𝐵))
6 0ov 7451 . . . 4 (𝐾𝐵) = ∅
75, 6eqtrdi 2784 . . 3 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
81, 7nsyl2 141 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9 fveq2 6891 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
109, 2eqtr4di 2786 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
1110oveqd 7431 . . . . . . 7 (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
1211eleq2d 2815 . . . . . 6 (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13 fvex 6904 . . . . . . . . 9 (mDV‘𝑡) ∈ V
1413elpw2 5341 . . . . . . . 8 (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15 fveq2 6891 . . . . . . . . . 10 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16 mclsval.d . . . . . . . . . 10 𝐷 = (mDV‘𝑇)
1715, 16eqtr4di 2786 . . . . . . . . 9 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
1817sseq2d 4010 . . . . . . . 8 (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾𝐷))
1914, 18bitrid 283 . . . . . . 7 (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾𝐷))
20 fvex 6904 . . . . . . . . 9 (mEx‘𝑡) ∈ V
2120elpw2 5341 . . . . . . . 8 (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22 fveq2 6891 . . . . . . . . . 10 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23 mclsval.e . . . . . . . . . 10 𝐸 = (mEx‘𝑇)
2422, 23eqtr4di 2786 . . . . . . . . 9 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
2524sseq2d 4010 . . . . . . . 8 (𝑡 = 𝑇 → (𝐵 ⊆ (mEx‘𝑡) ↔ 𝐵𝐸))
2621, 25bitrid 283 . . . . . . 7 (𝑡 = 𝑇 → (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵𝐸))
2719, 26anbi12d 631 . . . . . 6 (𝑡 = 𝑇 → ((𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)) ↔ (𝐾𝐷𝐵𝐸)))
2812, 27imbi12d 344 . . . . 5 (𝑡 = 𝑇 → ((𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡))) ↔ (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))))
29 vex 3474 . . . . . . 7 𝑡 ∈ V
3013pwex 5374 . . . . . . . 8 𝒫 (mDV‘𝑡) ∈ V
3120pwex 5374 . . . . . . . 8 𝒫 (mEx‘𝑡) ∈ V
3230, 31mpoex 8078 . . . . . . 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 35101 . . . . . . . 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 7010 . . . . . . 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 691 . . . . . 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 7656 . . . . 5 (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
3728, 36vtoclg 3539 . . . 4 (𝑇 ∈ V → (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸)))
388, 37mpcom 38 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))
3938simpld 494 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐾𝐷)
4038simprd 495 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐵𝐸)
418, 39, 403jca 1126 1 (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾𝐷𝐵𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085  wal 1532   = wceq 1534  wcel 2099  {cab 2705  wral 3057  Vcvv 3470  cun 3943  wss 3945  c0 4318  𝒫 cpw 4598  cotp 4632   cint 4944   class class class wbr 5142   × cxp 5670  ran crn 5673  cima 5675  cfv 6542  (class class class)co 7414  cmpo 7416  mAxcmax 35069  mExcmex 35071  mDVcmdv 35072  mVarscmvrs 35073  mSubstcmsub 35075  mVHcmvh 35076  mClscmcls 35081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-mcls 35101
This theorem is referenced by: (None)
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