Nuprl Lemma : restriction-of-transitive

[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ].  (Trans(T;x,y.R y)  Trans(T;x,y.R|P y))


Proof




Definitions occuring in Statement :  rel-restriction: R|P trans: Trans(T;x,y.E[x; y]) uall: [x:A]. B[x] prop: implies:  Q apply: a function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] implies:  Q member: t ∈ T prop: so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] trans: Trans(T;x,y.E[x; y]) rel-restriction: R|P all: x:A. B[x] and: P ∧ Q
Lemmas referenced :  trans_wf and_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality hypothesis functionEquality cumulativity universeEquality dependent_functionElimination productElimination independent_pairFormation independent_functionElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].    (Trans(T;x,y.R  x  y)  {}\mRightarrow{}  Trans(T;x,y.R|P  x  y))



Date html generated: 2016_05_14-AM-06_06_05
Last ObjectModification: 2015_12_26-AM-11_32_26

Theory : relations


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