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Theorem mpst123 34198
Description: Decompose a pre-statement into a triple of values. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
mpstssv.p 𝑃 = (mPreStβ€˜π‘‡)
Assertion
Ref Expression
mpst123 (𝑋 ∈ 𝑃 β†’ 𝑋 = ⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹)), (2nd β€˜π‘‹)⟩)

Proof of Theorem mpst123
StepHypRef Expression
1 mpstssv.p . . . 4 𝑃 = (mPreStβ€˜π‘‡)
21mpstssv 34197 . . 3 𝑃 βŠ† ((V Γ— V) Γ— V)
32sseli 3944 . 2 (𝑋 ∈ 𝑃 β†’ 𝑋 ∈ ((V Γ— V) Γ— V))
4 1st2nd2 7964 . . . 4 (𝑋 ∈ ((V Γ— V) Γ— V) β†’ 𝑋 = ⟨(1st β€˜π‘‹), (2nd β€˜π‘‹)⟩)
5 xp1st 7957 . . . . . 6 (𝑋 ∈ ((V Γ— V) Γ— V) β†’ (1st β€˜π‘‹) ∈ (V Γ— V))
6 1st2nd2 7964 . . . . . 6 ((1st β€˜π‘‹) ∈ (V Γ— V) β†’ (1st β€˜π‘‹) = ⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹))⟩)
75, 6syl 17 . . . . 5 (𝑋 ∈ ((V Γ— V) Γ— V) β†’ (1st β€˜π‘‹) = ⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹))⟩)
87opeq1d 4840 . . . 4 (𝑋 ∈ ((V Γ— V) Γ— V) β†’ ⟨(1st β€˜π‘‹), (2nd β€˜π‘‹)⟩ = ⟨⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹))⟩, (2nd β€˜π‘‹)⟩)
94, 8eqtrd 2773 . . 3 (𝑋 ∈ ((V Γ— V) Γ— V) β†’ 𝑋 = ⟨⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹))⟩, (2nd β€˜π‘‹)⟩)
10 df-ot 4599 . . 3 ⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹)), (2nd β€˜π‘‹)⟩ = ⟨⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹))⟩, (2nd β€˜π‘‹)⟩
119, 10eqtr4di 2791 . 2 (𝑋 ∈ ((V Γ— V) Γ— V) β†’ 𝑋 = ⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹)), (2nd β€˜π‘‹)⟩)
123, 11syl 17 1 (𝑋 ∈ 𝑃 β†’ 𝑋 = ⟨(1st β€˜(1st β€˜π‘‹)), (2nd β€˜(1st β€˜π‘‹)), (2nd β€˜π‘‹)⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1542   ∈ wcel 2107  Vcvv 3447  βŸ¨cop 4596  βŸ¨cotp 4598   Γ— cxp 5635  β€˜cfv 6500  1st c1st 7923  2nd c2nd 7924  mPreStcmpst 34131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-iota 6452  df-fun 6502  df-fv 6508  df-1st 7925  df-2nd 7926  df-mpst 34151
This theorem is referenced by:  msrf  34200  msrid  34203  mthmpps  34240
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