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Theorem mpstrcl 32795
Description: The elements of a pre-statement are sets. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mpstssv.p 𝑃 = (mPreSt‘𝑇)
Assertion
Ref Expression
mpstrcl (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))

Proof of Theorem mpstrcl
StepHypRef Expression
1 df-ot 4549 . . 3 𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴
2 mpstssv.p . . . . 5 𝑃 = (mPreSt‘𝑇)
32mpstssv 32793 . . . 4 𝑃 ⊆ ((V × V) × V)
43sseli 3939 . . 3 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
51, 4eqeltrrid 2917 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
6 opelxp 5564 . . . 4 (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
76anbi1i 626 . . 3 ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V))
8 opelxp 5564 . . 3 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
9 df-3an 1086 . . 3 ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V))
107, 8, 93bitr4i 306 . 2 (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
115, 10sylib 221 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  Vcvv 3471  cop 4546  cotp 4548   × cxp 5526  cfv 6328  mPreStcmpst 32727
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-ot 4549  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-mpst 32747
This theorem is referenced by:  elmsta  32802  mclsax  32823
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