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Theorem mrsubffval 33182
Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mrsubffval.c 𝐶 = (mCN‘𝑇)
mrsubffval.v 𝑉 = (mVR‘𝑇)
mrsubffval.r 𝑅 = (mREx‘𝑇)
mrsubffval.s 𝑆 = (mRSubst‘𝑇)
mrsubffval.g 𝐺 = (freeMnd‘(𝐶𝑉))
Assertion
Ref Expression
mrsubffval (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
Distinct variable groups:   𝑒,𝑓,𝑣,𝐶   𝑅,𝑒,𝑓,𝑣   𝑒,𝐺,𝑓   𝑇,𝑒,𝑓,𝑣   𝑒,𝑉,𝑓,𝑣
Allowed substitution hints:   𝑆(𝑣,𝑒,𝑓)   𝐺(𝑣)   𝑊(𝑣,𝑒,𝑓)

Proof of Theorem mrsubffval
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 mrsubffval.s . 2 𝑆 = (mRSubst‘𝑇)
2 elex 3426 . . 3 (𝑇𝑊𝑇 ∈ V)
3 fveq2 6717 . . . . . . 7 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4 mrsubffval.r . . . . . . 7 𝑅 = (mREx‘𝑇)
53, 4eqtr4di 2796 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6 fveq2 6717 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7 mrsubffval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
86, 7eqtr4di 2796 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
95, 8oveq12d 7231 . . . . 5 (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅pm 𝑉))
10 fveq2 6717 . . . . . . . . . . 11 (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
11 mrsubffval.c . . . . . . . . . . 11 𝐶 = (mCN‘𝑇)
1210, 11eqtr4di 2796 . . . . . . . . . 10 (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
1312, 8uneq12d 4078 . . . . . . . . 9 (𝑡 = 𝑇 → ((mCN‘𝑡) ∪ (mVR‘𝑡)) = (𝐶𝑉))
1413fveq2d 6721 . . . . . . . 8 (𝑡 = 𝑇 → (freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) = (freeMnd‘(𝐶𝑉)))
15 mrsubffval.g . . . . . . . 8 𝐺 = (freeMnd‘(𝐶𝑉))
1614, 15eqtr4di 2796 . . . . . . 7 (𝑡 = 𝑇 → (freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) = 𝐺)
1713mpteq1d 5144 . . . . . . . 8 (𝑡 = 𝑇 → (𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)))
1817coeq1d 5730 . . . . . . 7 (𝑡 = 𝑇 → ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))
1916, 18oveq12d 7231 . . . . . 6 (𝑡 = 𝑇 → ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))
205, 19mpteq12dv 5140 . . . . 5 (𝑡 = 𝑇 → (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
219, 20mpteq12dv 5140 . . . 4 (𝑡 = 𝑇 → (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
22 df-mrsub 33165 . . . 4 mRSubst = (𝑡 ∈ V ↦ (𝑓 ∈ ((mREx‘𝑡) ↑pm (mVR‘𝑡)) ↦ (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
23 ovex 7246 . . . . 5 (𝑅pm 𝑉) ∈ V
2423mptex 7039 . . . 4 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))) ∈ V
2521, 22, 24fvmpt 6818 . . 3 (𝑇 ∈ V → (mRSubst‘𝑇) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
262, 25syl 17 . 2 (𝑇𝑊 → (mRSubst‘𝑇) = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
271, 26syl5eq 2790 1 (𝑇𝑊𝑆 = (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2110  Vcvv 3408  cun 3864  ifcif 4439  cmpt 5135  dom cdm 5551  ccom 5555  cfv 6380  (class class class)co 7213  pm cpm 8509  ⟨“cs1 14152   Σg cgsu 16945  freeMndcfrmd 18274  mCNcmcn 33135  mVRcmvar 33136  mRExcmrex 33141  mRSubstcmrsub 33145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-mrsub 33165
This theorem is referenced by:  mrsubfval  33183  mrsubff  33187
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