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Theorem msrid 34203
Description: The reduct of a statement is itself. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mstaval.r 𝑅 = (mStRedβ€˜π‘‡)
mstaval.s 𝑆 = (mStatβ€˜π‘‡)
Assertion
Ref Expression
msrid (𝑋 ∈ 𝑆 β†’ (π‘…β€˜π‘‹) = 𝑋)

Proof of Theorem msrid
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 eqid 2733 . . . . 5 (mPreStβ€˜π‘‡) = (mPreStβ€˜π‘‡)
2 mstaval.r . . . . 5 𝑅 = (mStRedβ€˜π‘‡)
31, 2msrf 34200 . . . 4 𝑅:(mPreStβ€˜π‘‡)⟢(mPreStβ€˜π‘‡)
4 ffn 6672 . . . 4 (𝑅:(mPreStβ€˜π‘‡)⟢(mPreStβ€˜π‘‡) β†’ 𝑅 Fn (mPreStβ€˜π‘‡))
5 fvelrnb 6907 . . . 4 (𝑅 Fn (mPreStβ€˜π‘‡) β†’ (𝑋 ∈ ran 𝑅 ↔ βˆƒπ‘  ∈ (mPreStβ€˜π‘‡)(π‘…β€˜π‘ ) = 𝑋))
63, 4, 5mp2b 10 . . 3 (𝑋 ∈ ran 𝑅 ↔ βˆƒπ‘  ∈ (mPreStβ€˜π‘‡)(π‘…β€˜π‘ ) = 𝑋)
71mpst123 34198 . . . . . . . . . . 11 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ 𝑠 = ⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
87fveq2d 6850 . . . . . . . . . 10 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜π‘ ) = (π‘…β€˜βŸ¨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩))
9 id 22 . . . . . . . . . . . 12 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ 𝑠 ∈ (mPreStβ€˜π‘‡))
107, 9eqeltrrd 2835 . . . . . . . . . . 11 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ ⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ (mPreStβ€˜π‘‡))
11 eqid 2733 . . . . . . . . . . . 12 (mVarsβ€˜π‘‡) = (mVarsβ€˜π‘‡)
12 eqid 2733 . . . . . . . . . . . 12 βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) = βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))
1311, 1, 2, 12msrval 34196 . . . . . . . . . . 11 (⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜βŸ¨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩) = ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
1410, 13syl 17 . . . . . . . . . 10 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜βŸ¨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩) = ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
158, 14eqtrd 2773 . . . . . . . . 9 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜π‘ ) = ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
163ffvelcdmi 7038 . . . . . . . . 9 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜π‘ ) ∈ (mPreStβ€˜π‘‡))
1715, 16eqeltrrd 2835 . . . . . . . 8 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ (mPreStβ€˜π‘‡))
1811, 1, 2, 12msrval 34196 . . . . . . . 8 (⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜βŸ¨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩) = ⟨(((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
1917, 18syl 17 . . . . . . 7 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜βŸ¨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩) = ⟨(((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
20 inass 4183 . . . . . . . . . 10 (((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = ((1st β€˜(1st β€˜π‘ )) ∩ ((βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))))
21 inidm 4182 . . . . . . . . . . 11 ((βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))
2221ineq2i 4173 . . . . . . . . . 10 ((1st β€˜(1st β€˜π‘ )) ∩ ((βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))))) = ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))))
2320, 22eqtri 2761 . . . . . . . . 9 (((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))))
2423a1i 11 . . . . . . . 8 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))))
2524oteq1d 4846 . . . . . . 7 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ ⟨(((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ = ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
2619, 25eqtrd 2773 . . . . . 6 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜βŸ¨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩) = ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
2715fveq2d 6850 . . . . . 6 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜(π‘…β€˜π‘ )) = (π‘…β€˜βŸ¨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩))
2826, 27, 153eqtr4d 2783 . . . . 5 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ (π‘…β€˜(π‘…β€˜π‘ )) = (π‘…β€˜π‘ ))
29 fveq2 6846 . . . . . 6 ((π‘…β€˜π‘ ) = 𝑋 β†’ (π‘…β€˜(π‘…β€˜π‘ )) = (π‘…β€˜π‘‹))
30 id 22 . . . . . 6 ((π‘…β€˜π‘ ) = 𝑋 β†’ (π‘…β€˜π‘ ) = 𝑋)
3129, 30eqeq12d 2749 . . . . 5 ((π‘…β€˜π‘ ) = 𝑋 β†’ ((π‘…β€˜(π‘…β€˜π‘ )) = (π‘…β€˜π‘ ) ↔ (π‘…β€˜π‘‹) = 𝑋))
3228, 31syl5ibcom 244 . . . 4 (𝑠 ∈ (mPreStβ€˜π‘‡) β†’ ((π‘…β€˜π‘ ) = 𝑋 β†’ (π‘…β€˜π‘‹) = 𝑋))
3332rexlimiv 3142 . . 3 (βˆƒπ‘  ∈ (mPreStβ€˜π‘‡)(π‘…β€˜π‘ ) = 𝑋 β†’ (π‘…β€˜π‘‹) = 𝑋)
346, 33sylbi 216 . 2 (𝑋 ∈ ran 𝑅 β†’ (π‘…β€˜π‘‹) = 𝑋)
35 mstaval.s . . 3 𝑆 = (mStatβ€˜π‘‡)
362, 35mstaval 34202 . 2 𝑆 = ran 𝑅
3734, 36eleq2s 2852 1 (𝑋 ∈ 𝑆 β†’ (π‘…β€˜π‘‹) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070   βˆͺ cun 3912   ∩ cin 3913  {csn 4590  βŸ¨cotp 4598  βˆͺ cuni 4869   Γ— cxp 5635  ran crn 5638   β€œ cima 5640   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  1st c1st 7923  2nd c2nd 7924  mVarscmvrs 34127  mPreStcmpst 34131  mStRedcmsr 34132  mStatcmsta 34133
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-rep 5246  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-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  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-iun 4960  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-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1st 7925  df-2nd 7926  df-mpst 34151  df-msr 34152  df-msta 34153
This theorem is referenced by:  elmsta  34206
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