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Theorem mclsrcl 30712
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 3879 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → ¬ (𝐾𝐶𝐵) = ∅)
2 mclsval.c . . . . . 6 𝐶 = (mCls‘𝑇)
3 fvprc 6097 . . . . . 6 𝑇 ∈ V → (mCls‘𝑇) = ∅)
42, 3syl5eq 2656 . . . . 5 𝑇 ∈ V → 𝐶 = ∅)
54oveqd 6566 . . . 4 𝑇 ∈ V → (𝐾𝐶𝐵) = (𝐾𝐵))
6 0ov 6580 . . . 4 (𝐾𝐵) = ∅
75, 6syl6eq 2660 . . 3 𝑇 ∈ V → (𝐾𝐶𝐵) = ∅)
81, 7nsyl2 141 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝑇 ∈ V)
9 fveq2 6103 . . . . . . . . 9 (𝑡 = 𝑇 → (mCls‘𝑡) = (mCls‘𝑇))
109, 2syl6eqr 2662 . . . . . . . 8 (𝑡 = 𝑇 → (mCls‘𝑡) = 𝐶)
1110oveqd 6566 . . . . . . 7 (𝑡 = 𝑇 → (𝐾(mCls‘𝑡)𝐵) = (𝐾𝐶𝐵))
1211eleq2d 2673 . . . . . 6 (𝑡 = 𝑇 → (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) ↔ 𝐴 ∈ (𝐾𝐶𝐵)))
13 fvex 6113 . . . . . . . . 9 (mDV‘𝑡) ∈ V
1413elpw2 4755 . . . . . . . 8 (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾 ⊆ (mDV‘𝑡))
15 fveq2 6103 . . . . . . . . . 10 (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
16 mclsval.d . . . . . . . . . 10 𝐷 = (mDV‘𝑇)
1715, 16syl6eqr 2662 . . . . . . . . 9 (𝑡 = 𝑇 → (mDV‘𝑡) = 𝐷)
1817sseq2d 3596 . . . . . . . 8 (𝑡 = 𝑇 → (𝐾 ⊆ (mDV‘𝑡) ↔ 𝐾𝐷))
1914, 18syl5bb 271 . . . . . . 7 (𝑡 = 𝑇 → (𝐾 ∈ 𝒫 (mDV‘𝑡) ↔ 𝐾𝐷))
20 fvex 6113 . . . . . . . . 9 (mEx‘𝑡) ∈ V
2120elpw2 4755 . . . . . . . 8 (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵 ⊆ (mEx‘𝑡))
22 fveq2 6103 . . . . . . . . . 10 (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
23 mclsval.e . . . . . . . . . 10 𝐸 = (mEx‘𝑇)
2422, 23syl6eqr 2662 . . . . . . . . 9 (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
2524sseq2d 3596 . . . . . . . 8 (𝑡 = 𝑇 → (𝐵 ⊆ (mEx‘𝑡) ↔ 𝐵𝐸))
2621, 25syl5bb 271 . . . . . . 7 (𝑡 = 𝑇 → (𝐵 ∈ 𝒫 (mEx‘𝑡) ↔ 𝐵𝐸))
2719, 26anbi12d 743 . . . . . 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 3176 . . . . . . 7 𝑡 ∈ V
3013pwex 4774 . . . . . . . 8 𝒫 (mDV‘𝑡) ∈ V
3120pwex 4774 . . . . . . . 8 𝒫 (mEx‘𝑡) ∈ V
3230, 31mpt2ex 7136 . . . . . . 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 30648 . . . . . . . 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 6200 . . . . . . 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 704 . . . . . 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 6774 . . . . 5 (𝐴 ∈ (𝐾(mCls‘𝑡)𝐵) → (𝐾 ∈ 𝒫 (mDV‘𝑡) ∧ 𝐵 ∈ 𝒫 (mEx‘𝑡)))
3728, 36vtoclg 3239 . . . 4 (𝑇 ∈ V → (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸)))
388, 37mpcom 37 . . 3 (𝐴 ∈ (𝐾𝐶𝐵) → (𝐾𝐷𝐵𝐸))
3938simpld 474 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐾𝐷)
4038simprd 478 . 2 (𝐴 ∈ (𝐾𝐶𝐵) → 𝐵𝐸)
418, 39, 403jca 1235 1 (𝐴 ∈ (𝐾𝐶𝐵) → (𝑇 ∈ V ∧ 𝐾𝐷𝐵𝐸))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3a 1031  wal 1473   = wceq 1475  wcel 1977  {cab 2596  wral 2896  Vcvv 3173  cun 3538  wss 3540  c0 3874  𝒫 cpw 4108  cotp 4133   cint 4410   class class class wbr 4583   × cxp 5036  ran crn 5039  cima 5041  cfv 5804  (class class class)co 6549  cmpt2 6551  mAxcmax 30616  mExcmex 30618  mDVcmdv 30619  mVarscmvrs 30620  mSubstcmsub 30622  mVHcmvh 30623  mClscmcls 30628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-mcls 30648
This theorem is referenced by: (None)
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