Theorem mrsubffval 34188

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 3464	. . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
3		fveq2 6847	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = (mREx‘𝑇))
4		mrsubffval.r	. . . . . . 7 ⊢ 𝑅 = (mREx‘𝑇)
5	3, 4	eqtr4di 2789	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mREx‘𝑡) = 𝑅)
6		fveq2 6847	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = (mVR‘𝑇))
7		mrsubffval.v	. . . . . . 7 ⊢ 𝑉 = (mVR‘𝑇)
8	6, 7	eqtr4di 2789	. . . . . 6 ⊢ (𝑡 = 𝑇 → (mVR‘𝑡) = 𝑉)
9	5, 8	oveq12d 7380	. . . . 5 ⊢ (𝑡 = 𝑇 → ((mREx‘𝑡) ↑pm (mVR‘𝑡)) = (𝑅 ↑pm 𝑉))
10		fveq2 6847	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑇 → (mCN‘𝑡) = (mCN‘𝑇))
11		mrsubffval.c	. . . . . . . . . . 11 ⊢ 𝐶 = (mCN‘𝑇)
12	10, 11	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑡 = 𝑇 → (mCN‘𝑡) = 𝐶)
13	12, 8	uneq12d 4129	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → ((mCN‘𝑡) ∪ (mVR‘𝑡)) = (𝐶 ∪ 𝑉))
14	13	fveq2d 6851	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) = (freeMnd‘(𝐶 ∪ 𝑉)))
15		mrsubffval.g	. . . . . . . 8 ⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉))
16	14, 15	eqtr4di 2789	. . . . . . 7 ⊢ (𝑡 = 𝑇 → (freeMnd‘((mCN‘𝑡) ∪ (mVR‘𝑡))) = 𝐺)
17	13	mpteq1d 5205	. . . . . . . 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 5822	. . . . . . 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 7380	. . . . . 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 5201	. . . . 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 5201	. . . 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 34171	. . . 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 7395	. . . . 5 ⊢ (𝑅 ↑pm 𝑉) ∈ V
24	23	mptex 7178	. . . 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 6953	. . 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	eqtrid 2783	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 1541   ∈ wcel 2106  Vcvv 3446   ∪ cun 3911  ifcif 4491   ↦ cmpt 5193  dom cdm 5638   ∘ ccom 5642  ‘cfv 6501  (class class class)co 7362   ↑_pm cpm 8773  ⟨“cs1 14495   Σ_g cgsu 17336  freeMndcfrmd 18671  mCNcmcn 34141  mVRcmvar 34142  mRExcmrex 34147  mRSubstcmrsub 34151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pr 5389

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  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 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-ov 7365  df-mrsub 34171

This theorem is referenced by:  mrsubfval  34189  mrsubff  34193