Theorem mpstval 35522

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 6858	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
3		mpstval.v	. . . . . . . . 9 ⊢ 𝑉 = (mDV‘𝑇)
4	2, 3	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝑉)
5	4	pweqd 4580	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝑉)
6	5	rabeqdv 3421	. . . . . 6 ⊢ (𝑡 = 𝑇 → {𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑})
7		fveq2 6858	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
8		mpstval.e	. . . . . . . . 9 ⊢ 𝐸 = (mEx‘𝑇)
9	7, 8	eqtr4di 2782	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
10	9	pweqd 4580	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
11	10	ineq1d 4182	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝒫 (mEx‘𝑡) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
12	6, 11	xpeq12d 5669	. . . . 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 5669	. . . 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 35480	. . . 4 ⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
15	3	fvexi 6872	. . . . . . . 8 ⊢ 𝑉 ∈ V
16	15	pwex 5335	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
17	16	rabex 5294	. . . . . 6 ⊢ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∈ V
18	8	fvexi 6872	. . . . . . . 8 ⊢ 𝐸 ∈ V
19	18	pwex 5335	. . . . . . 7 ⊢ 𝒫 𝐸 ∈ V
20	19	inex1 5272	. . . . . 6 ⊢ (𝒫 𝐸 ∩ Fin) ∈ V
21	17, 20	xpex 7729	. . . . 5 ⊢ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∈ V
22	21, 18	xpex 7729	. . . 4 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ∈ V
23	13, 14, 22	fvmpt 6968	. . 3 ⊢ (𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
24		xp0 6131	. . . . 5 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅) = ∅
25	24	eqcomi 2738	. . . 4 ⊢ ∅ = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅)
26		fvprc 6850	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
27		fvprc 6850	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
28	8, 27	eqtrid 2776	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
29	28	xpeq2d 5668	. . . 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 2790	. . 3 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
31	23, 30	pm2.61i 182	. 2 ⊢ (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
32	1, 31	eqtri 2752	1 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  {crab 3405  Vcvv 3447   ∩ cin 3913  ∅c0 4296  𝒫 cpw 4563   × cxp 5636  ^◡ccnv 5637  ‘cfv 6511  Fincfn 8918  mExcmex 35454  mDVcmdv 35455  mPreStcmpst 35460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-mpst 35480

This theorem is referenced by:  elmpst  35523  mpstssv  35526