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Theorem elmpst 31884
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 5315 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴𝐸))
2 opelxp 5315 . . . . 5 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)))
3 cnveq 5466 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
4 id 22 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
53, 4eqeq12d 2780 . . . . . . . 8 (𝑑 = 𝐷 → (𝑑 = 𝑑𝐷 = 𝐷))
65elrab 3521 . . . . . . 7 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ↔ (𝐷 ∈ 𝒫 𝑉𝐷 = 𝐷))
7 mpstval.v . . . . . . . . . 10 𝑉 = (mDV‘𝑇)
87fvexi 6391 . . . . . . . . 9 𝑉 ∈ V
98elpw2 4988 . . . . . . . 8 (𝐷 ∈ 𝒫 𝑉𝐷𝑉)
109anbi1i 617 . . . . . . 7 ((𝐷 ∈ 𝒫 𝑉𝐷 = 𝐷) ↔ (𝐷𝑉𝐷 = 𝐷))
116, 10bitri 266 . . . . . 6 (𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ↔ (𝐷𝑉𝐷 = 𝐷))
12 elfpw 8477 . . . . . 6 (𝐻 ∈ (𝒫 𝐸 ∩ Fin) ↔ (𝐻𝐸𝐻 ∈ Fin))
1311, 12anbi12i 620 . . . . 5 ((𝐷 ∈ {𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} ∧ 𝐻 ∈ (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)))
142, 13bitri 266 . . . 4 (⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)))
1514anbi1i 617 . . 3 ((⟨𝐷, 𝐻⟩ ∈ ({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) ∧ 𝐴𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
161, 15bitri 266 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
17 df-ot 4345 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
18 mpstval.e . . . 4 𝐸 = (mEx‘𝑇)
19 mpstval.p . . . 4 𝑃 = (mPreSt‘𝑇)
207, 18, 19mpstval 31883 . . 3 𝑃 = (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸)
2117, 20eleq12i 2837 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ (({𝑑 ∈ 𝒫 𝑉𝑑 = 𝑑} × (𝒫 𝐸 ∩ Fin)) × 𝐸))
22 df-3an 1109 . 2 (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸) ↔ (((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin)) ∧ 𝐴𝐸))
2316, 21, 223bitr4i 294 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷𝑉𝐷 = 𝐷) ∧ (𝐻𝐸𝐻 ∈ Fin) ∧ 𝐴𝐸))
Colors of variables: wff setvar class
Syntax hints:  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {crab 3059  cin 3733  wss 3734  𝒫 cpw 4317  cop 4342  cotp 4344   × cxp 5277  ccnv 5278  cfv 6070  Fincfn 8162  mExcmex 31815  mDVcmdv 31816  mPreStcmpst 31821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-ot 4345  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-iota 6033  df-fun 6072  df-fv 6078  df-mpst 31841
This theorem is referenced by:  msrval  31886  msrf  31890  mclsssvlem  31910  mclsax  31917  mclsind  31918  mthmpps  31930  mclsppslem  31931  mclspps  31932
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