Nuprl Lemma : subtype_rel_dep_product_iff

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

Proof

Definitions occuring in Statement :  uiff: uiff(P;Q) uimplies: supposing a subtype_rel: A ⊆B uall: [x:A]. B[x] so_apply: x[s] function: x:A ⟶ B[x] product: x:A × B[x] universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a uiff: uiff(P;Q) and: P ∧ Q so_lambda: λ2x.t[x] so_apply: x[s] all: x:A. B[x] subtype_rel: A ⊆B prop: pi1: fst(t) pi2: snd(t)
Lemmas referenced :  subtype_rel_product uall_wf subtype_rel_wf pi2_wf pi1_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality sqequalRule lambdaEquality applyEquality independent_isectElimination hypothesis lambdaFormation because_Cache axiomEquality isect_memberEquality productEquality productElimination independent_pairEquality equalityTransitivity equalitySymmetry functionEquality cumulativity universeEquality dependent_pairEquality

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

Date html generated: 2016_05_13-PM-03_18_43
Last ObjectModification: 2015_12_26-AM-09_08_27

Theory : subtype_0

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