Nuprl Lemma : singlevalued-graph-iff
∀A:coSet{i:l}. ∀B:{a:coSet{i:l}| (a ∈ A)} ⟶ coSet{i:l}.
((∀a1,a2:coSet{i:l}. ((a1 ∈ A) ⇒ (a2 ∈ A) ⇒ seteq(a1;a2) ⇒ seteq(B[a1];B[a2])))
⇒ (∀x:coSet{i:l}
(singlevalued-graph(A;a.B[a];x)
⇐⇒ ∀a,b1,b2:coSet{i:l}.
((a ∈ A) ⇒ (b1 ∈ B[a]) ⇒ (b2 ∈ B[a]) ⇒ ((a,b1) ∈ x) ⇒ ((a,b2) ∈ x) ⇒ seteq(b1;b2)))))
Proof
Definitions occuring in Statement :
singlevalued-graph: singlevalued-graph(A;a.B[a];grph),
orderedpairset: (a,b),
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 :
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
rev_implies: P ⇐ Q,
singlevalued-graph: singlevalued-graph(A;a.B[a];grph),
set-predicate: set-predicate{i:l}(s;a.P[a]),
allsetmem: ∀a∈A.P[a],
guard: {T}
Lemmas referenced :
setmem_wf,
orderedpairset_wf,
singlevalued-graph_wf,
seteq_wf,
coSet_wf,
allsetmem-iff,
allsetmem_wf,
setmem_functionality_1,
orderedpairset_functionality,
seteq_weakening,
seteq_functionality,
set-item_wf,
set-dom_wf,
setmem_functionality,
implies-allsetmem
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
independent_pairFormation,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
dependent_set_memberEquality_alt,
inhabitedIsType,
dependent_functionElimination,
sqequalRule,
lambdaEquality_alt,
setIsType,
functionIsType,
because_Cache,
functionEquality,
setElimination,
rename,
independent_functionElimination,
productElimination
Latex:
\mforall{}A:coSet\{i:l\}. \mforall{}B:\{a:coSet\{i:l\}| (a \mmember{} A)\} {}\mrightarrow{} 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{} (\mforall{}x:coSet\{i:l\}
(singlevalued-graph(A;a.B[a];x)
\mLeftarrow{}{}\mRightarrow{} \mforall{}a,b1,b2:coSet\{i:l\}.
((a \mmember{} A)
{}\mRightarrow{} (b1 \mmember{} B[a])
{}\mRightarrow{} (b2 \mmember{} B[a])
{}\mRightarrow{} ((a,b1) \mmember{} x)
{}\mRightarrow{} ((a,b2) \mmember{} x)
{}\mRightarrow{} seteq(b1;b2)))))
Date html generated:
2020_05_20-PM-01_19_07
Last ObjectModification:
2020_01_06-PM-01_24_20
Theory : constructive!set!theory
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