Nuprl Lemma : sub-set_wf
∀[s:coSet{i:l}]. ∀[P:{a:coSet{i:l}| (a ∈ s)} ⟶ ℙ]. ({a ∈ s | P[a]} ∈ coSet{i:l})
Proof
Definitions occuring in Statement :
sub-set: {a ∈ s | P[a]},
setmem: (x ∈ s),
coSet: coSet{i:l},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
pi1: fst(t),
all: ∀x:A. B[x],
prop: ℙ,
so_apply: x[s],
mk-coset: mk-coset(T;f),
sub-set: {a ∈ s | P[a]},
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
setmem-coset,
setmem_wf,
coSet_wf,
mk-coset_wf,
coSet_subtype,
subtype_coSet
Rules used in proof :
functionEquality,
dependent_set_memberEquality,
dependent_functionElimination,
universeEquality,
lambdaEquality,
because_Cache,
setEquality,
functionExtensionality,
cumulativity,
productEquality,
isectElimination,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
applyEquality,
hypothesisEquality,
hypothesis,
extract_by_obid,
introduction,
hypothesis_subsumption,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[s:coSet\{i:l\}]. \mforall{}[P:\{a:coSet\{i:l\}| (a \mmember{} s)\} {}\mrightarrow{} \mBbbP{}]. (\{a \mmember{} s | P[a]\} \mmember{} coSet\{i:l\})
Date html generated:
2018_07_29-AM-09_52_20
Last ObjectModification:
2018_07_18-PM-04_50_09
Theory : constructive!set!theory
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