Nuprl Lemma : subtype_rel_dep_function

[A:Type]. ∀[B:A ⟶ Type]. ∀[C:Type]. ∀[D:C ⟶ Type].
  ((a:A ⟶ B[a]) ⊆(c:C ⟶ D[c])) supposing ((∀a:C. (B[a] ⊆D[a])) and (C ⊆A))


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

\mforall{}[A:Type].  \mforall{}[B:A  {}\mrightarrow{}  Type].  \mforall{}[C:Type].  \mforall{}[D:C  {}\mrightarrow{}  Type].
    ((a:A  {}\mrightarrow{}  B[a])  \msubseteq{}r  (c:C  {}\mrightarrow{}  D[c]))  supposing  ((\mforall{}a:C.  (B[a]  \msubseteq{}r  D[a]))  and  (C  \msubseteq{}r  A))

Date html generated: 2016_05_13-PM-03_18_39
Last ObjectModification: 2015_12_26-AM-09_08_26

Theory : subtype_0

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