Nuprl Lemma : tunion_subtype_base
∀[A:Type]. ∀[B:A ⟶ Type]. ⋃a:A.B[a] ⊆r Base supposing ∀a:A. (B[a] ⊆r Base)
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],
base: Base,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
tunion: ⋃x:A.B[x],
subtype_rel: A ⊆r B,
pi2: snd(t),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x]
Lemmas referenced :
base_wf,
subtype_rel_wf,
all_wf,
image-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
lambdaEquality,
imageElimination,
productElimination,
thin,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
isectElimination,
productEquality,
hypothesisEquality,
applyEquality,
baseClosed,
axiomEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[B:A {}\mrightarrow{} Type]. \mcup{}a:A.B[a] \msubseteq{}r Base supposing \mforall{}a:A. (B[a] \msubseteq{}r Base)
Date html generated:
2016_05_13-PM-03_19_31
Last ObjectModification:
2016_01_14-PM-04_32_01
Theory : subtype_0
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