### Nuprl Lemma : subtype_rel_isect_as_subtyping_lemma

`∀[A,T:Type]. ∀[B:T ⟶ Type].  A ⊆r (⋂x:T. B[x]) supposing ∀[x:T]. (A ⊆r B[x])`

Proof

Definitions occuring in Statement :  uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` so_apply: `x[s]` isect: `⋂x:A. B[x]` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  uall_wf subtype_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaEquality isect_memberEquality hypothesisEquality applyEquality hypothesis sqequalHypSubstitution isectElimination thin sqequalRule axiomEquality lemma_by_obid because_Cache equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality

Latex:
\mforall{}[A,T:Type].  \mforall{}[B:T  {}\mrightarrow{}  Type].    A  \msubseteq{}r  (\mcap{}x:T.  B[x])  supposing  \mforall{}[x:T].  (A  \msubseteq{}r  B[x])

Date html generated: 2016_05_13-PM-03_18_53
Last ObjectModification: 2015_12_26-AM-09_08_07

Theory : subtype_0

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