Theorem elmpst 35541

Description: Property of being a pre-statement. (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
elmpst	⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸))

Proof of Theorem elmpst

Dummy variable 𝑑 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opelxp 5721	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴 ∈ 𝐸))
2		opelxp 5721	. . . . 5 ⊢ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3		cnveq 5884	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → ◡𝑑 = ◡𝐷)
4		id 22	. . . . . . . . 9 ⊢ (𝑑 = 𝐷 → 𝑑 = 𝐷)
5	3, 4	eqeq12d 2753	. . . . . . . 8 ⊢ (𝑑 = 𝐷 → (◡𝑑 = 𝑑 ↔ ◡𝐷 = 𝐷))
6	5	elrab 3692	. . . . . . 7 ⊢ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉 ∧ ◡𝐷 = 𝐷))
7		mpstval.v	. . . . . . . . . 10 ⊢ 𝑉 = (mDV‘𝑇)
8	7	fvexi 6920	. . . . . . . . 9 ⊢ 𝑉 ∈ V
9	8	elpw2 5334	. . . . . . . 8 ⊢ (𝐷 ∈ 𝒫 𝑉 ↔ 𝐷 ⊆ 𝑉)
10	9	anbi1i 624	. . . . . . 7 ⊢ ((𝐷 ∈ 𝒫 𝑉 ∧ ◡𝐷 = 𝐷) ↔ (𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷))
11	6, 10	bitri 275	. . . . . 6 ⊢ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ↔ (𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷))
12		elfpw 9394	. . . . . 6 ⊢ (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin))
13	11, 12	anbi12i 628	. . . . 5 ⊢ ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)))
14	2, 13	bitri 275	. . . 4 ⊢ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)))
15	14	anbi1i 624	. . 3 ⊢ ((⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴 ∈ 𝐸) ↔ (((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
16	1, 15	bitri 275	. 2 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
17		df-ot 4635	. . 3 ⊢ ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
18		mpstval.e	. . . 4 ⊢ 𝐸 = (mEx‘𝑇)
19		mpstval.p	. . . 4 ⊢ 𝑃 = (mPreSt‘𝑇)
20	7, 18, 19	mpstval 35540	. . 3 ⊢ 𝑃 = (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
21	17, 20	eleq12i 2834	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉 ∣ ◡𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
22		df-3an 1089	. 2 ⊢ (((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸) ↔ (((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin)) ∧ 𝐴 ∈ 𝐸))
23	16, 21, 22	3bitr4i 303	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 ⊆ 𝑉 ∧ ◡𝐷 = 𝐷) ∧ (𝐻 ⊆ 𝐸 ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ 𝐸))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108  {crab 3436   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ⟨cop 4632  ⟨cotp 4634   × cxp 5683  ^◡ccnv 5684  ‘cfv 6561  Fincfn 8985  mExcmex 35472  mDVcmdv 35473  mPreStcmpst 35478

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-mpst 35498

This theorem is referenced by:  msrval  35543  msrf  35547  mclsssvlem  35567  mclsax  35574  mclsind  35575  mthmpps  35587  mclsppslem  35588  mclspps  35589