Nuprl Lemma : Piset_wf2
∀[A:Set{i:l}]. ∀[B:{a:Set{i:l}| (a ∈ A)} ⟶ Set{i:l}]. (Πa:A.B[a] ∈ Set{i:l})
Proof
Definitions occuring in Statement :
Piset: Πa:A.B[a],
Set: Set{i:l},
setmem: (x ∈ s),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
exists: ∃x:A. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
Piset: Πa:A.B[a],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
set-subtype-coSet,
piset_wf,
subtype_rel_set,
coSet_wf,
coSet-mem-Set-implies-Set,
singlevalued-graph_wf,
setmem_wf,
Set_wf,
piset_wf2,
sub-set_wf2
Rules used in proof :
isect_memberEquality,
functionEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
instantiate,
dependent_pairFormation,
independent_isectElimination,
dependent_set_memberEquality,
rename,
setElimination,
dependent_functionElimination,
because_Cache,
cumulativity,
hypothesis,
setEquality,
applyEquality,
lambdaEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:Set\{i:l\}]. \mforall{}[B:\{a:Set\{i:l\}| (a \mmember{} A)\} {}\mrightarrow{} Set\{i:l\}]. (\mPi{}a:A.B[a] \mmember{} Set\{i:l\})
Date html generated:
2018_07_29-AM-10_05_08
Last ObjectModification:
2018_07_18-PM-03_32_59
Theory : constructive!set!theory
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