Theorem mpstrcl 30692

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

Step	Hyp	Ref	Expression
1		df-ot 4134	. . 3 ⊢ ⟨𝐷, 𝐻, 𝐴⟩ = ⟨⟨𝐷, 𝐻⟩, 𝐴⟩
2		mpstssv.p	. . . . 5 ⊢ 𝑃 = (mPreSt‘𝑇)
3	2	mpstssv 30690	. . . 4 ⊢ 𝑃 ⊆ ((V × V) × V)
4	3	sseli 3564	. . 3 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨𝐷, 𝐻, 𝐴⟩ ∈ ((V × V) × V))
5	1, 4	syl5eqelr 2693	. 2 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → ⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V))
6		opelxp 5070	. . . 4 ⊢ (⟨𝐷, 𝐻⟩ ∈ (V × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V))
7	6	anbi1i 727	. . 3 ⊢ ((⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V))
8		opelxp 5070	. . 3 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (⟨𝐷, 𝐻⟩ ∈ (V × V) ∧ 𝐴 ∈ V))
9		df-3an 1033	. . 3 ⊢ ((𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V) ↔ ((𝐷 ∈ V ∧ 𝐻 ∈ V) ∧ 𝐴 ∈ V))
10	7, 8, 9	3bitr4i 291	. 2 ⊢ (⟨⟨𝐷, 𝐻⟩, 𝐴⟩ ∈ ((V × V) × V) ↔ (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))
11	5, 10	sylib 207	1 ⊢ (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 → (𝐷 ∈ V ∧ 𝐻 ∈ V ∧ 𝐴 ∈ V))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ⟨cotp 4133   × cxp 5036  ‘cfv 5804  mPreStcmpst 30624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-mpst 30644

This theorem is referenced by:  elmsta  30699  mclsax  30720