### Nuprl Lemma : restriction-of-transitive

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

Proof

Definitions occuring in Statement :  rel-restriction: `R|P` trans: `Trans(T;x,y.E[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x y.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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