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Theorem elmpst 34070
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 ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸))

Proof of Theorem elmpst
Dummy variable 𝑑 is distinct from all other variables.
StepHypRef Expression
1 opelxp 5669 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴𝐸))
2 opelxp 5669 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3 cnveq 5829 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
4 id 22 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
53, 4eqeq12d 2752 . . . . . . . 8 (𝑑 = 𝐷 → (𝑑 = 𝑑𝐷 = 𝐷))
65elrab 3645 . . . . . . 7 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉𝐷 = 𝐷))
7 mpstval.v . . . . . . . . . 10 𝑉 = (mDV‘𝑇)
87fvexi 6856 . . . . . . . . 9 𝑉 ∈ V
98elpw2 5302 . . . . . . . 8 (𝐷 ∈ 𝒫 𝑉𝐷𝑉)
109anbi1i 624 . . . . . . 7 ((𝐷 ∈ 𝒫 𝑉𝐷 = 𝐷) ↔ (𝐷𝑉𝐷 = 𝐷))
116, 10bitri 274 . . . . . 6 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ↔ (𝐷𝑉𝐷 = 𝐷))
12 elfpw 9297 . . . . . 6 (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻𝐸𝐻 ∈ Fin))
1311, 12anbi12i 627 . . . . 5 ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)))
142, 13bitri 274 . . . 4 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)))
1514anbi1i 624 . . 3 ((⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
161, 15bitri 274 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
17 df-ot 4595 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
18 mpstval.e . . . 4 𝐸 = (mEx‘𝑇)
19 mpstval.p . . . 4 𝑃 = (mPreSt‘𝑇)
207, 18, 19mpstval 34069 . . 3 𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
2117, 20eleq12i 2830 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
22 df-3an 1089 . 2 (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
2316, 21, 223bitr4i 302 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  {crab 3407  cin 3909  wss 3910  𝒫 cpw 4560  cop 4592  cotp 4594   × cxp 5631  ccnv 5632  cfv 6496  Fincfn 8882  mExcmex 34001  mDVcmdv 34002  mPreStcmpst 34007
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 7671
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-ot 4595  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 34027
This theorem is referenced by:  msrval  34072  msrf  34076  mclsssvlem  34096  mclsax  34103  mclsind  34104  mthmpps  34116  mclsppslem  34117  mclspps  34118
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