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Theorem mrsubffval 35492
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 3499 . . 3 (𝑇𝑊𝑇 ∈ V)
3 fveq2 6907 . . . . . . 7 (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4 mrsubffval.r . . . . . . 7 𝑅 = (mREx‘𝑇)
53, 4eqtr4di 2793 . . . . . 6 (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6 fveq2 6907 . . . . . . 7 (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7 mrsubffval.v . . . . . . 7 𝑉 = (mVR‘𝑇)
86, 7eqtr4di 2793 . . . . . 6 (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
95, 8oveq12d 7449 . . . . 5 (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅pm 𝑉))
10 fveq2 6907 . . . . . . . . . . 11 (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
11 mrsubffval.c . . . . . . . . . . 11 𝐶 = (mCN‘𝑇)
1210, 11eqtr4di 2793 . . . . . . . . . 10 (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
1312, 8uneq12d 4179 . . . . . . . . 9 (𝑡 = 𝑇 → ((mCN‘𝑡) ∪ (mVR‘𝑡)) = (𝐶𝑉))
1413fveq2d 6911 . . . . . . . 8 (𝑡 = 𝑇 → (freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) = (freeMnd‘(𝐶𝑉)))
15 mrsubffval.g . . . . . . . 8 𝐺 = (freeMnd‘(𝐶𝑉))
1614, 15eqtr4di 2793 . . . . . . 7 (𝑡 = 𝑇 → (freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) = 𝐺)
1713mpteq1d 5243 . . . . . . . 8 (𝑡 = 𝑇 → (𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)))
1817coeq1d 5875 . . . . . . 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 7449 . . . . . 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 5239 . . . . 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 5239 . . . 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 35475 . . . 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 7464 . . . . 5 (𝑅pm 𝑉) ∈ V
2423mptex 7243 . . . 4 (𝑓 ∈ (𝑅pm 𝑉) ↦ (𝑒𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))) ∈ V
2521, 22, 24fvmpt 7016 . . 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, 26eqtrid 2787 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 1537  wcel 2106  Vcvv 3478  cun 3961  ifcif 4531  cmpt 5231  dom cdm 5689  ccom 5693  cfv 6563  (class class class)co 7431  pm cpm 8866  ⟨“cs1 14630   Σg cgsu 17487  freeMndcfrmd 18873  mCNcmcn 35445  mVRcmvar 35446  mRExcmrex 35451  mRSubstcmrsub 35455
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-mrsub 35475
This theorem is referenced by:  mrsubfval  35493  mrsubff  35497
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