Nuprl Lemma : set-function_wf
∀[s:coSet{i:l}]. ∀[f:{x:coSet{i:l}| (x ∈ s)} ⟶ coSet{i:l}]. (set-function{i:l}(s; x.f[x]) ∈ ℙ')
Proof
Definitions occuring in Statement :
set-function: set-function{i:l}(s; x.f[x]),
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 :
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
set-function: set-function{i:l}(s; x.f[x]),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
seteq_wf,
setmem_wf,
coSet_wf,
all_wf
Rules used in proof :
isect_memberEquality,
setEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
because_Cache,
dependent_set_memberEquality,
applyEquality,
hypothesisEquality,
functionEquality,
cumulativity,
lambdaEquality,
hypothesis,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
instantiate,
thin,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[s:coSet\{i:l\}]. \mforall{}[f:\{x:coSet\{i:l\}| (x \mmember{} s)\} {}\mrightarrow{} coSet\{i:l\}]. (set-function\{i:l\}(s; x.f[x]) \mmember{} \mBbbP{}')
Date html generated:
2018_07_29-AM-09_52_33
Last ObjectModification:
2018_07_18-PM-02_27_21
Theory : constructive!set!theory
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