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Theorem elmpst 32857
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 5568 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴𝐸))
2 opelxp 5568 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3 cnveq 5721 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
4 id 22 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
53, 4eqeq12d 2838 . . . . . . . 8 (𝑑 = 𝐷 → (𝑑 = 𝑑𝐷 = 𝐷))
65elrab 3655 . . . . . . 7 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉𝐷 = 𝐷))
7 mpstval.v . . . . . . . . . 10 𝑉 = (mDV‘𝑇)
87fvexi 6666 . . . . . . . . 9 𝑉 ∈ V
98elpw2 5224 . . . . . . . 8 (𝐷 ∈ 𝒫 𝑉𝐷𝑉)
109anbi1i 626 . . . . . . 7 ((𝐷 ∈ 𝒫 𝑉𝐷 = 𝐷) ↔ (𝐷𝑉𝐷 = 𝐷))
116, 10bitri 278 . . . . . 6 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ↔ (𝐷𝑉𝐷 = 𝐷))
12 elfpw 8814 . . . . . 6 (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻𝐸𝐻 ∈ Fin))
1311, 12anbi12i 629 . . . . 5 ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)))
142, 13bitri 278 . . . 4 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)))
1514anbi1i 626 . . 3 ((⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
161, 15bitri 278 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
17 df-ot 4548 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
18 mpstval.e . . . 4 𝐸 = (mEx‘𝑇)
19 mpstval.p . . . 4 𝑃 = (mPreSt‘𝑇)
207, 18, 19mpstval 32856 . . 3 𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
2117, 20eleq12i 2906 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
22 df-3an 1086 . 2 (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
2316, 21, 223bitr4i 306 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2114  {crab 3134  cin 3907  wss 3908  𝒫 cpw 4511  cop 4545  cotp 4547   × cxp 5530  ccnv 5531  cfv 6334  Fincfn 8496  mExcmex 32788  mDVcmdv 32789  mPreStcmpst 32794
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-ot 4548  df-uni 4814  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-iota 6293  df-fun 6336  df-fv 6342  df-mpst 32814
This theorem is referenced by:  msrval  32859  msrf  32863  mclsssvlem  32883  mclsax  32890  mclsind  32891  mthmpps  32903  mclsppslem  32904  mclspps  32905
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