Theorem mpstval 32782

Description: A pre-statement is an ordered triple, whose first member is a symmetric set of disjoint variable conditions, whose second member is a finite set of expressions, and whose third member is an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
mpstval.v	⊢ 𝑉 = (mDV‘𝑇)
mpstval.e	⊢ 𝐸 = (mEx‘𝑇)
mpstval.p	⊢ 𝑃 = (mPreSt‘𝑇)

Assertion

Ref	Expression
mpstval	⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)

Distinct variable groups:   𝑇,𝑑   𝑉,𝑑

Allowed substitution hints:   𝑃(𝑑)   𝐸(𝑑)

Proof of Theorem mpstval

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mpstval.p	. 2 ⊢ 𝑃 = (mPreSt‘𝑇)
2		fveq2 6669	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
3		mpstval.v	. . . . . . . . 9 ⊢ 𝑉 = (mDV‘𝑇)
4	2, 3	syl6eqr 2874	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝑉)
5	4	pweqd 4557	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝑉)
6	5	rabeqdv 3484	. . . . . 6 ⊢ (𝑡 = 𝑇 → {𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑})
7		fveq2 6669	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
8		mpstval.e	. . . . . . . . 9 ⊢ 𝐸 = (mEx‘𝑇)
9	7, 8	syl6eqr 2874	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
10	9	pweqd 4557	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
11	10	ineq1d 4187	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝒫 (mEx‘𝑡) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
12	6, 11	xpeq12d 5585	. . . . 5 ⊢ (𝑡 = 𝑇 → ({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) = ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)))
13	12, 9	xpeq12d 5585	. . . 4 ⊢ (𝑡 = 𝑇 → (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
14		df-mpst 32740	. . . 4 ⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
15	3	fvexi 6683	. . . . . . . 8 ⊢ 𝑉 ∈ V
16	15	pwex 5280	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
17	16	rabex 5234	. . . . . 6 ⊢ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∈ V
18	8	fvexi 6683	. . . . . . . 8 ⊢ 𝐸 ∈ V
19	18	pwex 5280	. . . . . . 7 ⊢ 𝒫 𝐸 ∈ V
20	19	inex1 5220	. . . . . 6 ⊢ (𝒫 𝐸 ∩ Fin) ∈ V
21	17, 20	xpex 7475	. . . . 5 ⊢ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∈ V
22	21, 18	xpex 7475	. . . 4 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ∈ V
23	13, 14, 22	fvmpt 6767	. . 3 ⊢ (𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
24		xp0 6014	. . . . 5 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅) = ∅
25	24	eqcomi 2830	. . . 4 ⊢ ∅ = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅)
26		fvprc 6662	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
27		fvprc 6662	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
28	8, 27	syl5eq 2868	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
29	28	xpeq2d 5584	. . . 4 ⊢ (¬ 𝑇 ∈ V → (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅))
30	25, 26, 29	3eqtr4a 2882	. . 3 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
31	23, 30	pm2.61i 184	. 2 ⊢ (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
32	1, 31	eqtri 2844	1 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2110  {crab 3142  Vcvv 3494   ∩ cin 3934  ∅c0 4290  𝒫 cpw 4538   × cxp 5552  ^◡ccnv 5553  ‘cfv 6354  Fincfn 8508  mExcmex 32714  mDVcmdv 32715  mPreStcmpst 32720

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-mpst 32740

This theorem is referenced by:  elmpst  32783  mpstssv  32786