Nuprl Lemma : subtype_rel_tunion
∀[A:Type]. ∀[B:A ⟶ Type]. ∀[C:Type]. ∀[D:C ⟶ Type].
(⋃a:A.B[a] ⊆r ⋃c:C.D[c]) supposing ((∀a:A. (B[a] ⊆r D[a])) and (A ⊆r C))
Proof
Definitions occuring in Statement :
uimplies: b supposing a,
subtype_rel: A ⊆r B,
tunion: ⋃x:A.B[x],
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 :
tunion: ⋃x:A.B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
subtype_rel_wf,
all_wf,
subtype_rel_product,
subtype_rel_image
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
applyEquality,
baseClosed,
independent_isectElimination,
lambdaEquality,
hypothesis,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[C:Type]. \mforall{}[D:C {}\mrightarrow{} Type].
(\mcup{}a:A.B[a] \msubseteq{}r \mcup{}c:C.D[c]) supposing ((\mforall{}a:A. (B[a] \msubseteq{}r D[a])) and (A \msubseteq{}r C))
Date html generated:
2016_05_13-PM-03_18_41
Last ObjectModification:
2016_01_14-PM-04_32_01
Theory : subtype_0
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