Nuprl Lemma : restriction-to-field

[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ].  ((∀x,y:T.  ((R y)  ((P x) ∧ (P y))))  (∀x,y:T.  (R|P ⇐⇒ y)))


Proof




Definitions occuring in Statement :  rel-restriction: R|P uall: [x:A]. B[x] prop: all: x:A. B[x] iff: ⇐⇒ Q implies:  Q and: P ∧ Q apply: a function: x:A ⟶ B[x] universe: Type
Definitions unfolded in proof :  rel-restriction: R|P uall: [x:A]. B[x] implies:  Q all: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q member: t ∈ T prop: rev_implies:  Q so_lambda: λ2x.t[x] so_apply: x[s] guard: {T}
Lemmas referenced :  and_wf all_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin hypothesis cut lemma_by_obid isectElimination applyEquality hypothesisEquality lambdaEquality functionEquality cumulativity universeEquality dependent_functionElimination independent_functionElimination

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



Date html generated: 2016_05_14-AM-06_06_08
Last ObjectModification: 2015_12_26-AM-11_32_35

Theory : relations


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