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Theorem mpst123 31946
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 31945 . . 3 𝑃 ⊆ ((V × V) × V)
32sseli 3792 . 2 (𝑋𝑃𝑋 ∈ ((V × V) × V))
4 1st2nd2 7438 . . . 4 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨(1st𝑋), (2nd𝑋)⟩)
5 xp1st 7431 . . . . . 6 (𝑋 ∈ ((V × V) × V) → (1st𝑋) ∈ (V × V))
6 1st2nd2 7438 . . . . . 6 ((1st𝑋) ∈ (V × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
75, 6syl 17 . . . . 5 (𝑋 ∈ ((V × V) × V) → (1st𝑋) = ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩)
87opeq1d 4597 . . . 4 (𝑋 ∈ ((V × V) × V) → ⟨(1st𝑋), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
94, 8eqtrd 2831 . . 3 (𝑋 ∈ ((V × V) × V) → 𝑋 = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩)
10 df-ot 4375 . . 3 ⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋)), (2nd𝑋)⟩ = ⟨⟨(1st ‘(1st𝑋)), (2nd ‘(1st𝑋))⟩, (2nd𝑋)⟩
119, 10syl6eqr 2849 . 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 1653  wcel 2157  Vcvv 3383  cop 4372  cotp 4374   × cxp 5308  cfv 6099  1st c1st 7397  2nd c2nd 7398  mPreStcmpst 31879
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-ot 4375  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fv 6107  df-1st 7399  df-2nd 7400  df-mpst 31899
This theorem is referenced by:  msrf  31948  msrid  31951  mthmpps  31988
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