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Theorem mpst123 31765
 Description: Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mpstssv.p 𝑃 = (mPreSt‘𝑇)
Assertion
Ref Expression
mpst123 (𝑋𝑃𝑋 = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩)

Proof of Theorem mpst123
StepHypRef Expression
1 mpstssv.p . . . 4 𝑃 = (mPreSt‘𝑇)
21mpstssv 31764 . . 3 𝑃 ⊆ ((V × V) × V)
32sseli 3740 . 2 (𝑋𝑃𝑋 ∈ ((V × V) × V))
4 1st2nd2 7373 . . . 4 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 xp1st 7366 . . . . . 6 (𝑋 ∈ ((V × V) × V) → (1st𝑋) ∈ (V × V))
6 1st2nd2 7373 . . . . . 6 ((1st𝑋) ∈ (V × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
75, 6syl 17 . . . . 5 (𝑋 ∈ ((V × V) × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
87opeq1d 4559 . . . 4 (𝑋 ∈ ((V × V) × V) → ⟨(1st𝑋), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
94, 8eqtrd 2794 . . 3 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
10 df-ot 4330 . . 3 ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩
119, 10syl6eqr 2812 . 2 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩)
123, 11syl 17 1 (𝑋𝑃𝑋 = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  Vcvv 3340  ⟨cop 4327  ⟨cotp 4329   × cxp 5264  ‘cfv 6049  1st c1st 7332  2nd c2nd 7333  mPreStcmpst 31698 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-ot 4330  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fv 6057  df-1st 7334  df-2nd 7335  df-mpst 31718 This theorem is referenced by:  msrf  31767  msrid  31770  mthmpps  31807
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