Nuprl Lemma : member-fset-image
∀[T,A:Type]. ∀[eqt:EqDecider(T)]. ∀[eqa:EqDecider(A)]. ∀[f:T ⟶ A]. ∀[s:fset(T)]. ∀[x:T]. f x ∈ f"(s) supposing x ∈ s
Proof
Definitions occuring in Statement :
fset-image: f"(s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
exists: ∃x:A. B[x],
prop: ℙ,
squash: ↓T,
implies: P ⇒ Q
Lemmas referenced :
deq_wf,
fset_wf,
fset-image_wf,
fset-member_witness,
equal_wf,
fset-member_wf,
and_wf,
member-fset-image-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
applyEquality,
productElimination,
independent_isectElimination,
dependent_pairFormation,
independent_pairFormation,
hypothesis,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_functionElimination,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[eqt:EqDecider(T)]. \mforall{}[eqa:EqDecider(A)]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[s:fset(T)]. \mforall{}[x:T].
f x \mmember{} f"(s) supposing x \mmember{} s
Date html generated:
2016_05_14-PM-03_44_01
Last ObjectModification:
2016_01_14-PM-10_40_30
Theory : finite!sets
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