Nuprl Lemma : setmem-mk-coset
∀[T,f,x:Top]. ((x ∈ mk-coset(T;f)) ~ ∃t:T. seteq(x;f t))
Proof
Definitions occuring in Statement :
setmem: (x ∈ s),
seteq: seteq(s1;s2),
mk-coset: mk-coset(T;f),
uall: ∀[x:A]. B[x],
top: Top,
exists: ∃x:A. B[x],
apply: f a,
sqequal: s ~ t
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
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),
seteq: seteq(s1;s2)
Lemmas referenced :
top_wf
Rules used in proof :
because_Cache,
hypothesisEquality,
thin,
isectElimination,
isect_memberEquality,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
sqequalAxiom,
cut,
introduction,
isect_memberFormation,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution
Latex:
\mforall{}[T,f,x:Top]. ((x \mmember{} mk-coset(T;f)) \msim{} \mexists{}t:T. seteq(x;f t))
Date html generated:
2018_07_29-AM-09_50_07
Last ObjectModification:
2018_07_18-AM-10_53_51
Theory : constructive!set!theory
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