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Theorem mrsubffval 34793
Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubffval.c 𝐢 = (mCNβ€˜π‘‡)
mrsubffval.v 𝑉 = (mVRβ€˜π‘‡)
mrsubffval.r 𝑅 = (mRExβ€˜π‘‡)
mrsubffval.s 𝑆 = (mRSubstβ€˜π‘‡)
mrsubffval.g 𝐺 = (freeMndβ€˜(𝐢 βˆͺ 𝑉))
Assertion
Ref Expression
mrsubffval (𝑇 ∈ π‘Š β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
Distinct variable groups:   𝑒,𝑓,𝑣,𝐢   𝑅,𝑒,𝑓,𝑣   𝑒,𝐺,𝑓   𝑇,𝑒,𝑓,𝑣   𝑒,𝑉,𝑓,𝑣
Allowed substitution hints:   𝑆(𝑣,𝑒,𝑓)   𝐺(𝑣)   π‘Š(𝑣,𝑒,𝑓)

Proof of Theorem mrsubffval
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.s . 2 𝑆 = (mRSubstβ€˜π‘‡)
2 elex 3492 . . 3 (𝑇 ∈ π‘Š β†’ 𝑇 ∈ V)
3 fveq2 6892 . . . . . . 7 (𝑑 = 𝑇 β†’ (mRExβ€˜π‘‘) = (mRExβ€˜π‘‡))
4 mrsubffval.r . . . . . . 7 𝑅 = (mRExβ€˜π‘‡)
53, 4eqtr4di 2789 . . . . . 6 (𝑑 = 𝑇 β†’ (mRExβ€˜π‘‘) = 𝑅)
6 fveq2 6892 . . . . . . 7 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = (mVRβ€˜π‘‡))
7 mrsubffval.v . . . . . . 7 𝑉 = (mVRβ€˜π‘‡)
86, 7eqtr4di 2789 . . . . . 6 (𝑑 = 𝑇 β†’ (mVRβ€˜π‘‘) = 𝑉)
95, 8oveq12d 7430 . . . . 5 (𝑑 = 𝑇 β†’ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) = (𝑅 ↑pm 𝑉))
10 fveq2 6892 . . . . . . . . . . 11 (𝑑 = 𝑇 β†’ (mCNβ€˜π‘‘) = (mCNβ€˜π‘‡))
11 mrsubffval.c . . . . . . . . . . 11 𝐢 = (mCNβ€˜π‘‡)
1210, 11eqtr4di 2789 . . . . . . . . . 10 (𝑑 = 𝑇 β†’ (mCNβ€˜π‘‘) = 𝐢)
1312, 8uneq12d 4165 . . . . . . . . 9 (𝑑 = 𝑇 β†’ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) = (𝐢 βˆͺ 𝑉))
1413fveq2d 6896 . . . . . . . 8 (𝑑 = 𝑇 β†’ (freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) = (freeMndβ€˜(𝐢 βˆͺ 𝑉)))
15 mrsubffval.g . . . . . . . 8 𝐺 = (freeMndβ€˜(𝐢 βˆͺ 𝑉))
1614, 15eqtr4di 2789 . . . . . . 7 (𝑑 = 𝑇 β†’ (freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) = 𝐺)
1713mpteq1d 5244 . . . . . . . 8 (𝑑 = 𝑇 β†’ (𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) = (𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)))
1817coeq1d 5862 . . . . . . 7 (𝑑 = 𝑇 β†’ ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒) = ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))
1916, 18oveq12d 7430 . . . . . 6 (𝑑 = 𝑇 β†’ ((freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) Ξ£g ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)) = (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))
205, 19mpteq12dv 5240 . . . . 5 (𝑑 = 𝑇 β†’ (𝑒 ∈ (mRExβ€˜π‘‘) ↦ ((freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) Ξ£g ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))) = (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒))))
219, 20mpteq12dv 5240 . . . 4 (𝑑 = 𝑇 β†’ (𝑓 ∈ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) ↦ (𝑒 ∈ (mRExβ€˜π‘‘) ↦ ((freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) Ξ£g ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
22 df-mrsub 34776 . . . 4 mRSubst = (𝑑 ∈ V ↦ (𝑓 ∈ ((mRExβ€˜π‘‘) ↑pm (mVRβ€˜π‘‘)) ↦ (𝑒 ∈ (mRExβ€˜π‘‘) ↦ ((freeMndβ€˜((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘))) Ξ£g ((𝑣 ∈ ((mCNβ€˜π‘‘) βˆͺ (mVRβ€˜π‘‘)) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
23 ovex 7445 . . . . 5 (𝑅 ↑pm 𝑉) ∈ V
2423mptex 7228 . . . 4 (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))) ∈ V
2521, 22, 24fvmpt 6999 . . 3 (𝑇 ∈ V β†’ (mRSubstβ€˜π‘‡) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
262, 25syl 17 . 2 (𝑇 ∈ π‘Š β†’ (mRSubstβ€˜π‘‡) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
271, 26eqtrid 2783 1 (𝑇 ∈ π‘Š β†’ 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Ξ£g ((𝑣 ∈ (𝐢 βˆͺ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (π‘“β€˜π‘£), βŸ¨β€œπ‘£β€βŸ©)) ∘ 𝑒)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1540   ∈ wcel 2105  Vcvv 3473   βˆͺ cun 3947  ifcif 4529   ↦ cmpt 5232  dom cdm 5677   ∘ ccom 5681  β€˜cfv 6544  (class class class)co 7412   ↑pm cpm 8824  βŸ¨β€œcs1 14550   Ξ£g cgsu 17391  freeMndcfrmd 18765  mCNcmcn 34746  mVRcmvar 34747  mRExcmrex 34752  mRSubstcmrsub 34756
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  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-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 7415  df-mrsub 34776
This theorem is referenced by:  mrsubfval  34794  mrsubff  34798
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