Nuprl Lemma : setmem-funset
∀A,B,x:coSet{i:l}. ((x ∈ A ⟶ B) ⇐⇒ function-graph{i:l}(A;_.B;x))
Proof
Definitions occuring in Statement :
funset: A ⟶ B,
function-graph: function-graph{i:l}(A;a.B[a];grph),
setmem: (x ∈ s),
coSet: coSet{i:l},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
funset: A ⟶ B,
all: ∀x:A. B[x]
Lemmas referenced :
iff_wf,
Piset_wf,
seteq_wf,
seteq_weakening,
setmem-Piset,
setmem_wf,
coSet_wf,
function-graph_wf
Rules used in proof :
instantiate,
independent_functionElimination,
dependent_functionElimination,
impliesFunctionality,
productElimination,
addLevel,
because_Cache,
cumulativity,
setEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}A,B,x:coSet\{i:l\}. ((x \mmember{} A {}\mrightarrow{} B) \mLeftarrow{}{}\mRightarrow{} function-graph\{i:l\}(A;$_{}$.B;x))
Date html generated:
2018_07_29-AM-10_05_29
Last ObjectModification:
2018_07_18-PM-04_39_54
Theory : constructive!set!theory
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