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‘(𝐶 ∪ 𝑉))

Assertion

Ref	Expression
mrsubffval	⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“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.

Step	Hyp	Ref	Expression
1		mrsubffval.s	. 2 ⊢ 𝑆 = (mRSubst‘𝑇)
2		elex 3426	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
3		fveq2 6717	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4		mrsubffval.r	. . . . . . 7 ⊢ 𝑅 = (mREx‘𝑇)
5	3, 4	eqtr4di 2796	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6		fveq2 6717	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7		mrsubffval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
8	6, 7	eqtr4di 2796	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
9	5, 8	oveq12d 7231	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅 ↑pm 𝑉))
10		fveq2 6717	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
11		mrsubffval.c	. . . . . . . . . . 11 ⊢ 𝐶 = (mCN‘𝑇)
12	10, 11	eqtr4di 2796	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
13	12, 8	uneq12d 4078	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((mCN‘𝑡) ∪ (mVR‘𝑡)) = (𝐶 ∪ 𝑉))
14	13	fveq2d 6721	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) = (freeMnd‘(𝐶 ∪ 𝑉)))
15		mrsubffval.g	. . . . . . . 8 ⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉))
16	14, 15	eqtr4di 2796	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) = 𝐺)
17	13	mpteq1d 5144	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)))
18	17	coeq1d 5730	. . . . . . 7 ⊢ (𝑡 = 𝑇 → ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))
19	16, 18	oveq12d 7231	. . . . . 6 ⊢ (𝑡 = 𝑇 → ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))
20	5, 19	mpteq12dv 5140	. . . . 5 ⊢ (𝑡 = 𝑇 → (𝑒 ∈ (mREx‘𝑡) ↦ ((freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) Σg ((𝑣 ∈ ((mCN‘𝑡) ∪ (mVR‘𝑡)) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
21	9, 20	mpteq12dv 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, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“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, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
23		ovex 7246	. . . . 5 ⊢ (𝑅 ↑pm 𝑉) ∈ V
24	23	mptex 7039	. . . 4 ⊢ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))) ∈ V
25	21, 22, 24	fvmpt 6818	. . 3 ⊢ (𝑇 ∈ V → (mRSubst‘𝑇) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
26	2, 25	syl 17	. 2 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇) = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
27	1, 26	syl5eq 2790	1 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))

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   Σ_g 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