### Nuprl Lemma : subtype_rel_isect-2

`∀[A:Type]. ∀[B1,B2:A ⟶ Type].  (⋂x:A. B1[x]) ⊆r (⋂x:A. B2[x]) supposing ∀[x:A]. (B1[x] ⊆r B2[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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` and: `P ∧ Q` subtype_rel: `A ⊆r B` all: `∀x:A. B[x]` prop: `ℙ`
Lemmas referenced :  subtype_rel_isect_general uall_wf subtype_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality because_Cache sqequalRule lambdaEquality applyEquality independent_isectElimination independent_pairFormation lambdaFormation hypothesis axiomEquality isect_memberEquality equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality

Latex:
\mforall{}[A:Type].  \mforall{}[B1,B2:A  {}\mrightarrow{}  Type].    (\mcap{}x:A.  B1[x])  \msubseteq{}r  (\mcap{}x:A.  B2[x])  supposing  \mforall{}[x:A].  (B1[x]  \msubseteq{}r  B2[x])

Date html generated: 2016_05_13-PM-03_18_54
Last ObjectModification: 2015_12_26-AM-09_08_06

Theory : subtype_0

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