Theorem mpstval 34129

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 6842	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = (mDV‘𝑇))
3		mpstval.v	. . . . . . . . 9 ⊢ 𝑉 = (mDV‘𝑇)
4	2, 3	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mDV‘𝑡) = 𝑉)
5	4	pweqd 4577	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mDV‘𝑡) = 𝒫 𝑉)
6	5	rabeqdv 3422	. . . . . 6 ⊢ (𝑡 = 𝑇 → {𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} = {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑})
7		fveq2 6842	. . . . . . . . 9 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = (mEx‘𝑇))
8		mpstval.e	. . . . . . . . 9 ⊢ 𝐸 = (mEx‘𝑇)
9	7, 8	eqtr4di 2794	. . . . . . . 8 ⊢ (𝑡 = 𝑇 → (mEx‘𝑡) = 𝐸)
10	9	pweqd 4577	. . . . . . 7 ⊢ (𝑡 = 𝑇 → 𝒫 (mEx‘𝑡) = 𝒫 𝐸)
11	10	ineq1d 4171	. . . . . 6 ⊢ (𝑡 = 𝑇 → (𝒫 (mEx‘𝑡) ∩ Fin) = (𝒫 𝐸 ∩ Fin))
12	6, 11	xpeq12d 5664	. . . . 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 5664	. . . 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 34087	. . . 4 ⊢ mPreSt = (𝑡 ∈ V ↦ (({𝑑 ∈ 𝒫 (mDV‘𝑡) ∣ ◡𝑑 = 𝑑} × (𝒫 (mEx‘𝑡) ∩ Fin)) × (mEx‘𝑡)))
15	3	fvexi 6856	. . . . . . . 8 ⊢ 𝑉 ∈ V
16	15	pwex 5335	. . . . . . 7 ⊢ 𝒫 𝑉 ∈ V
17	16	rabex 5289	. . . . . 6 ⊢ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∈ V
18	8	fvexi 6856	. . . . . . . 8 ⊢ 𝐸 ∈ V
19	18	pwex 5335	. . . . . . 7 ⊢ 𝒫 𝐸 ∈ V
20	19	inex1 5274	. . . . . 6 ⊢ (𝒫 𝐸 ∩ Fin) ∈ V
21	17, 20	xpex 7687	. . . . 5 ⊢ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∈ V
22	21, 18	xpex 7687	. . . 4 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ∈ V
23	13, 14, 22	fvmpt 6948	. . 3 ⊢ (𝑇 ∈ V → (mPreSt‘𝑇) = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
24		xp0 6110	. . . . 5 ⊢ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅) = ∅
25	24	eqcomi 2745	. . . 4 ⊢ ∅ = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × ∅)
26		fvprc 6834	. . . 4 ⊢ (¬ 𝑇 ∈ V → (mPreSt‘𝑇) = ∅)
27		fvprc 6834	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
28	8, 27	eqtrid 2788	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
29	28	xpeq2d 5663	. . . 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 2802	. . 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 2764	1 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2106  {crab 3407  Vcvv 3445   ∩ cin 3909  ∅c0 4282  𝒫 cpw 4560   × cxp 5631  ^◡ccnv 5632  ‘cfv 6496  Fincfn 8883  mExcmex 34061  mDVcmdv 34062  mPreStcmpst 34067

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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672

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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-mpst 34087

This theorem is referenced by:  elmpst  34130  mpstssv  34133