Nuprl Lemma : mem-mk-set_wf
∀[T:Type]. ∀[f:T ⟶ coSet{i:l}]. ∀[t:T]. (mem-mk-set(f;t) ∈ (f t ∈ mk-coset(T;f)))
Proof
Definitions occuring in Statement :
mem-mk-set: mem-mk-set(f;t),
setmem: (x ∈ s),
mk-coset: mk-coset(T;f),
coSet: coSet{i:l},
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
exists: ∃x:A. B[x],
seteq: seteq(s1;s2),
pi2: snd(t),
pi1: fst(t),
coW-dom: coW-dom(a.B[a];w),
coW-item: coW-item(w;b),
coWmem: coWmem(a.B[a];z;w),
setmem: (x ∈ s),
mk-coset: mk-coset(T;f),
mem-mk-set: mem-mk-set(f;t),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
seteqweaken_wf,
equal_wf,
coSet_wf,
seteq_wf
Rules used in proof :
dependent_functionElimination,
instantiate,
universeEquality,
functionEquality,
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
hypothesis,
because_Cache,
cumulativity,
functionExtensionality,
applyEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesisEquality,
dependent_pairEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} coSet\{i:l\}]. \mforall{}[t:T]. (mem-mk-set(f;t) \mmember{} (f t \mmember{} mk-coset(T;f)))
Date html generated:
2018_07_29-AM-10_08_25
Last ObjectModification:
2018_07_18-PM-00_28_23
Theory : constructive!set!theory
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