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Theorem mclsrcl 34552
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 4334 . . 3 (𝐴 ∈ (𝐾𝐢𝐡) β†’ Β¬ (𝐾𝐢𝐡) = βˆ…)
2 mclsval.c . . . . . 6 𝐢 = (mClsβ€˜π‘‡)
3 fvprc 6884 . . . . . 6 (Β¬ 𝑇 ∈ V β†’ (mClsβ€˜π‘‡) = βˆ…)
42, 3eqtrid 2785 . . . . 5 (Β¬ 𝑇 ∈ V β†’ 𝐢 = βˆ…)
54oveqd 7426 . . . 4 (Β¬ 𝑇 ∈ V β†’ (𝐾𝐢𝐡) = (πΎβˆ…π΅))
6 0ov 7446 . . . 4 (πΎβˆ…π΅) = βˆ…
75, 6eqtrdi 2789 . . 3 (Β¬ 𝑇 ∈ V β†’ (𝐾𝐢𝐡) = βˆ…)
81, 7nsyl2 141 . 2 (𝐴 ∈ (𝐾𝐢𝐡) β†’ 𝑇 ∈ V)
9 fveq2 6892 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mClsβ€˜π‘‘) = (mClsβ€˜π‘‡))
109, 2eqtr4di 2791 . . . . . . . 8 (𝑑 = 𝑇 β†’ (mClsβ€˜π‘‘) = 𝐢)
1110oveqd 7426 . . . . . . 7 (𝑑 = 𝑇 β†’ (𝐾(mClsβ€˜π‘‘)𝐡) = (𝐾𝐢𝐡))
1211eleq2d 2820 . . . . . 6 (𝑑 = 𝑇 β†’ (𝐴 ∈ (𝐾(mClsβ€˜π‘‘)𝐡) ↔ 𝐴 ∈ (𝐾𝐢𝐡)))
13 fvex 6905 . . . . . . . . 9 (mDVβ€˜π‘‘) ∈ V
1413elpw2 5346 . . . . . . . 8 (𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ↔ 𝐾 βŠ† (mDVβ€˜π‘‘))
15 fveq2 6892 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (mDVβ€˜π‘‘) = (mDVβ€˜π‘‡))
16 mclsval.d . . . . . . . . . 10 𝐷 = (mDVβ€˜π‘‡)
1715, 16eqtr4di 2791 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mDVβ€˜π‘‘) = 𝐷)
1817sseq2d 4015 . . . . . . . 8 (𝑑 = 𝑇 β†’ (𝐾 βŠ† (mDVβ€˜π‘‘) ↔ 𝐾 βŠ† 𝐷))
1914, 18bitrid 283 . . . . . . 7 (𝑑 = 𝑇 β†’ (𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ↔ 𝐾 βŠ† 𝐷))
20 fvex 6905 . . . . . . . . 9 (mExβ€˜π‘‘) ∈ V
2120elpw2 5346 . . . . . . . 8 (𝐡 ∈ 𝒫 (mExβ€˜π‘‘) ↔ 𝐡 βŠ† (mExβ€˜π‘‘))
22 fveq2 6892 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = (mExβ€˜π‘‡))
23 mclsval.e . . . . . . . . . 10 𝐸 = (mExβ€˜π‘‡)
2422, 23eqtr4di 2791 . . . . . . . . 9 (𝑑 = 𝑇 β†’ (mExβ€˜π‘‘) = 𝐸)
2524sseq2d 4015 . . . . . . . 8 (𝑑 = 𝑇 β†’ (𝐡 βŠ† (mExβ€˜π‘‘) ↔ 𝐡 βŠ† 𝐸))
2621, 25bitrid 283 . . . . . . 7 (𝑑 = 𝑇 β†’ (𝐡 ∈ 𝒫 (mExβ€˜π‘‘) ↔ 𝐡 βŠ† 𝐸))
2719, 26anbi12d 632 . . . . . 6 (𝑑 = 𝑇 β†’ ((𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ∧ 𝐡 ∈ 𝒫 (mExβ€˜π‘‘)) ↔ (𝐾 βŠ† 𝐷 ∧ 𝐡 βŠ† 𝐸)))
2812, 27imbi12d 345 . . . . 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 3479 . . . . . . 7 𝑑 ∈ V
3013pwex 5379 . . . . . . . 8 𝒫 (mDVβ€˜π‘‘) ∈ V
3120pwex 5379 . . . . . . . 8 𝒫 (mExβ€˜π‘‘) ∈ V
3230, 31mpoex 8066 . . . . . . 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 34488 . . . . . . . 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 7010 . . . . . . 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 691 . . . . . 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 7648 . . . . 5 (𝐴 ∈ (𝐾(mClsβ€˜π‘‘)𝐡) β†’ (𝐾 ∈ 𝒫 (mDVβ€˜π‘‘) ∧ 𝐡 ∈ 𝒫 (mExβ€˜π‘‘)))
3728, 36vtoclg 3557 . . . 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 496 . 2 (𝐴 ∈ (𝐾𝐢𝐡) β†’ 𝐾 βŠ† 𝐷)
4038simprd 497 . 2 (𝐴 ∈ (𝐾𝐢𝐡) β†’ 𝐡 βŠ† 𝐸)
418, 39, 403jca 1129 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 397   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3062  Vcvv 3475   βˆͺ cun 3947   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βŸ¨cotp 4637  βˆ© cint 4951   class class class wbr 5149   Γ— cxp 5675  ran crn 5678   β€œ cima 5680  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  mAxcmax 34456  mExcmex 34458  mDVcmdv 34459  mVarscmvrs 34460  mSubstcmsub 34462  mVHcmvh 34463  mClscmcls 34468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-mcls 34488
This theorem is referenced by: (None)
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