Nuprl Lemma : setmem-Piset
∀A:coSet{i:l}. ∀B:{a:coSet{i:l}| (a ∈ A)} ⟶ coSet{i:l}. ∀x:coSet{i:l}.
((∀a1,a2:coSet{i:l}. ((a1 ∈ A) ⇒ (a2 ∈ A) ⇒ seteq(a1;a2) ⇒ seteq(B[a1];B[a2])))
⇒ ((x ∈ Πa:A.B[a]) ⇐⇒ function-graph{i:l}(A;a.B[a];x)))
Proof
Definitions occuring in Statement :
function-graph: function-graph{i:l}(A;a.B[a];grph),
Piset: Πa:A.B[a],
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
guard: {T},
subtype_rel: A ⊆r B,
cand: A c∧ B,
function-graph: function-graph{i:l}(A;a.B[a];grph),
set-predicate: set-predicate{i:l}(s;a.P[a]),
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
Piset: Πa:A.B[a],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
implies-setmem-piset,
Set_wf,
set-subtype-coSet,
orderedpairset_wf,
singlevalued-graph-iff,
setmem-piset-implies,
all_wf,
iff_wf,
sub-set_wf,
seteq_wf,
singlevalued-graph_functionality,
setmem-sub-coset,
function-graph_wf,
singlevalued-graph_wf,
coSet_wf,
piset_wf,
setmem_wf
Rules used in proof :
because_Cache,
dependent_set_memberEquality,
functionEquality,
instantiate,
independent_functionElimination,
rename,
setElimination,
impliesFunctionality,
productElimination,
addLevel,
dependent_functionElimination,
cumulativity,
hypothesis,
setEquality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
productEquality,
independent_pairFormation,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}A:coSet\{i:l\}. \mforall{}B:\{a:coSet\{i:l\}| (a \mmember{} A)\} {}\mrightarrow{} coSet\{i:l\}. \mforall{}x:coSet\{i:l\}.
((\mforall{}a1,a2:coSet\{i:l\}. ((a1 \mmember{} A) {}\mRightarrow{} (a2 \mmember{} A) {}\mRightarrow{} seteq(a1;a2) {}\mRightarrow{} seteq(B[a1];B[a2])))
{}\mRightarrow{} ((x \mmember{} \mPi{}a:A.B[a]) \mLeftarrow{}{}\mRightarrow{} function-graph\{i:l\}(A;a.B[a];x)))
Date html generated:
2018_07_29-AM-10_05_19
Last ObjectModification:
2018_07_18-PM-04_38_54
Theory : constructive!set!theory
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