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Theorem mclsrcl 34850
Description: Reverse closure for the closure function. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mclsval.d 𝐷 = (mDVβ€˜π‘‡)
mclsval.e 𝐸 = (mExβ€˜π‘‡)
mclsval.c 𝐢 = (mClsβ€˜π‘‡)
Assertion
Ref Expression
mclsrcl (𝐴 ∈ (𝐾𝐢𝐡) β†’ (𝑇 ∈ V ∧ 𝐾 βŠ† 𝐷 ∧ 𝐡 βŠ† 𝐸))

Proof of Theorem mclsrcl
Dummy variables β„Ž 𝑑 𝑑 𝑐 π‘š π‘œ 𝑝 𝑠 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0i 4332 . . 3 (𝐴 ∈ (𝐾𝐢𝐡) β†’ Β¬ (𝐾𝐢𝐡) = βˆ…)
2 mclsval.c . . . . . 6 𝐢 = (mClsβ€˜π‘‡)
3 fvprc 6882 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mClsβ€˜π‘‡) = βˆ…)
42, 3eqtrid 2782 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝐢 = βˆ…)
54oveqd 7428 . . . 4 (Β¬ 𝑇 ∈ V β†’ (𝐾𝐢𝐡) = (πΎβˆ…π΅))
6 0ov 7448 . . . 4 (πΎβˆ…π΅) = βˆ…
75, 6eqtrdi 2786 . . 3 (Β¬ 𝑇 ∈ V β†’ (𝐾𝐢𝐡) = βˆ…)
81, 7nsyl2 141 . 2 (𝐴 ∈ (𝐾𝐢𝐡) β†’ 𝑇 ∈ V)
9 fveq2 6890 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mClsβ€˜π‘‘) = (mClsβ€˜π‘‡))
109, 2eqtr4di 2788 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mClsβ€˜π‘‘) = 𝐢)
1110oveqd 7428 . . . . . . 7 (𝑑 = 𝑇 β†’ (𝐾(mClsβ€˜π‘‘)𝐡) = (𝐾𝐢𝐡))
1211eleq2d 2817 . . . . . 6 (𝑑 = 𝑇 β†’ (𝐴 ∈ (𝐾(mClsβ€˜π‘‘)𝐡) ↔ 𝐴 ∈ (𝐾𝐢𝐡)))
13 fvex 6903 . . . . . . . . 9 (mDVβ€˜π‘‘) ∈ V
1413elpw2 5344 . . . . . . . 8 (𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ↔ 𝐾 βŠ† (mDVβ€˜π‘‘))
15 fveq2 6890 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (mDVβ€˜π‘‘) = (mDVβ€˜π‘‡))
16 mclsval.d . . . . . . . . . 10 𝐷 = (mDVβ€˜π‘‡)
1715, 16eqtr4di 2788 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mDVβ€˜π‘‘) = 𝐷)
1817sseq2d 4013 . . . . . . . 8 (𝑑 = 𝑇 β†’ (𝐾 βŠ† (mDVβ€˜π‘‘) ↔ 𝐾 βŠ† 𝐷))
1914, 18bitrid 282 . . . . . . 7 (𝑑 = 𝑇 β†’ (𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ↔ 𝐾 βŠ† 𝐷))
20 fvex 6903 . . . . . . . . 9 (mExβ€˜π‘‘) ∈ V
2120elpw2 5344 . . . . . . . 8 (𝐡 ∈ 𝒫 (mExβ€˜π‘‘) ↔ 𝐡 βŠ† (mExβ€˜π‘‘))
22 fveq2 6890 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
23 mclsval.e . . . . . . . . . 10 𝐸 = (mExβ€˜π‘‡)
2422, 23eqtr4di 2788 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
2524sseq2d 4013 . . . . . . . 8 (𝑑 = 𝑇 β†’ (𝐡 βŠ† (mExβ€˜π‘‘) ↔ 𝐡 βŠ† 𝐸))
2621, 25bitrid 282 . . . . . . 7 (𝑑 = 𝑇 β†’ (𝐡 ∈ 𝒫 (mExβ€˜π‘‘) ↔ 𝐡 βŠ† 𝐸))
2719, 26anbi12d 629 . . . . . 6 (𝑑 = 𝑇 β†’ ((𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ∧ 𝐡 ∈ 𝒫 (mExβ€˜π‘‘)) ↔ (𝐾 βŠ† 𝐷 ∧ 𝐡 βŠ† 𝐸)))
2812, 27imbi12d 343 . . . . 5 (𝑑 = 𝑇 β†’ ((𝐴 ∈ (𝐾(mClsβ€˜π‘‘)𝐡) β†’ (𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ∧ 𝐡 ∈ 𝒫 (mExβ€˜π‘‘))) ↔ (𝐴 ∈ (𝐾𝐢𝐡) β†’ (𝐾 βŠ† 𝐷 ∧ 𝐡 βŠ† 𝐸))))
29 vex 3476 . . . . . . 7 𝑑 ∈ V
3013pwex 5377 . . . . . . . 8 𝒫 (mDVβ€˜π‘‘) ∈ V
3120pwex 5377 . . . . . . . 8 𝒫 (mExβ€˜π‘‘) ∈ V
3230, 31mpoex 8068 . . . . . . 7 (𝑑 ∈ 𝒫 (mDVβ€˜π‘‘), β„Ž ∈ 𝒫 (mExβ€˜π‘‘) ↦ ∩ {𝑐 ∣ ((β„Ž βˆͺ ran (mVHβ€˜π‘‘)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‘) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‘)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‘))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘¦)))) βŠ† 𝑑)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))}) ∈ V
33 df-mcls 34786 . . . . . . . 8 mCls = (𝑑 ∈ V ↦ (𝑑 ∈ 𝒫 (mDVβ€˜π‘‘), β„Ž ∈ 𝒫 (mExβ€˜π‘‘) ↦ ∩ {𝑐 ∣ ((β„Ž βˆͺ ran (mVHβ€˜π‘‘)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‘) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‘)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‘))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘¦)))) βŠ† 𝑑)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))}))
3433fvmpt2 7008 . . . . . . 7 ((𝑑 ∈ V ∧ (𝑑 ∈ 𝒫 (mDVβ€˜π‘‘), β„Ž ∈ 𝒫 (mExβ€˜π‘‘) ↦ ∩ {𝑐 ∣ ((β„Ž βˆͺ ran (mVHβ€˜π‘‘)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‘) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‘)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‘))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘¦)))) βŠ† 𝑑)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))}) ∈ V) β†’ (mClsβ€˜π‘‘) = (𝑑 ∈ 𝒫 (mDVβ€˜π‘‘), β„Ž ∈ 𝒫 (mExβ€˜π‘‘) ↦ ∩ {𝑐 ∣ ((β„Ž βˆͺ ran (mVHβ€˜π‘‘)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‘) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‘)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‘))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘¦)))) βŠ† 𝑑)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))}))
3529, 32, 34mp2an 688 . . . . . 6 (mClsβ€˜π‘‘) = (𝑑 ∈ 𝒫 (mDVβ€˜π‘‘), β„Ž ∈ 𝒫 (mExβ€˜π‘‘) ↦ ∩ {𝑐 ∣ ((β„Ž βˆͺ ran (mVHβ€˜π‘‘)) βŠ† 𝑐 ∧ βˆ€π‘šβˆ€π‘œβˆ€π‘(βŸ¨π‘š, π‘œ, π‘βŸ© ∈ (mAxβ€˜π‘‘) β†’ βˆ€π‘  ∈ ran (mSubstβ€˜π‘‘)(((𝑠 β€œ (π‘œ βˆͺ ran (mVHβ€˜π‘‘))) βŠ† 𝑐 ∧ βˆ€π‘₯βˆ€π‘¦(π‘₯π‘šπ‘¦ β†’ (((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘₯))) Γ— ((mVarsβ€˜π‘‘)β€˜(π‘ β€˜((mVHβ€˜π‘‘)β€˜π‘¦)))) βŠ† 𝑑)) β†’ (π‘ β€˜π‘) ∈ 𝑐)))})
3635elmpocl 7650 . . . . 5 (𝐴 ∈ (𝐾(mClsβ€˜π‘‘)𝐡) β†’ (𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ∧ 𝐡 ∈ 𝒫 (mExβ€˜π‘‘)))
3728, 36vtoclg 3541 . . . 4 (𝑇 ∈ V β†’ (𝐴 ∈ (𝐾𝐢𝐡) β†’ (𝐾 βŠ† 𝐷 ∧ 𝐡 βŠ† 𝐸)))
388, 37mpcom 38 . . 3 (𝐴 ∈ (𝐾𝐢𝐡) β†’ (𝐾 βŠ† 𝐷 ∧ 𝐡 βŠ† 𝐸))
3938simpld 493 . 2 (𝐴 ∈ (𝐾𝐢𝐡) β†’ 𝐾 βŠ† 𝐷)
4038simprd 494 . 2 (𝐴 ∈ (𝐾𝐢𝐡) β†’ 𝐡 βŠ† 𝐸)
418, 39, 403jca 1126 1 (𝐴 ∈ (𝐾𝐢𝐡) β†’ (𝑇 ∈ V ∧ 𝐾 βŠ† 𝐷 ∧ 𝐡 βŠ† 𝐸))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1085  βˆ€wal 1537   = wceq 1539   ∈ wcel 2104  {cab 2707  βˆ€wral 3059  Vcvv 3472   βˆͺ cun 3945   βŠ† wss 3947  βˆ…c0 4321  π’« cpw 4601  βŸ¨cotp 4635  βˆ© cint 4949   class class class wbr 5147   Γ— cxp 5673  ran crn 5676   β€œ cima 5678  β€˜cfv 6542  (class class class)co 7411   ∈ cmpo 7413  mAxcmax 34754  mExcmex 34756  mDVcmdv 34757  mVarscmvrs 34758  mSubstcmsub 34760  mVHcmvh 34761  mClscmcls 34766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  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 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-mcls 34786
This theorem is referenced by: (None)
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