Nuprl Lemma : singlevalued-graph_functionality
∀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])))
⇒ (∀g1,g2:coSet{i:l}. (seteq(g1;g2) ⇒ (singlevalued-graph(A;a.B[a];g1) ⇐⇒ singlevalued-graph(A;a.B[a];g2)))))
Proof
Definitions occuring in Statement :
singlevalued-graph: singlevalued-graph(A;a.B[a];grph),
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},
rev_implies: P ⇐ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
seteq_weakening,
setmem_functionality,
all_wf,
seteq_wf,
singlevalued-graph_wf,
orderedpairset_wf,
setmem_wf,
coSet_wf,
singlevalued-graph-iff
Rules used in proof :
functionEquality,
instantiate,
because_Cache,
dependent_set_memberEquality,
productElimination,
independent_functionElimination,
isectElimination,
cumulativity,
hypothesis,
setEquality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
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{}a1,a2:coSet\{i:l\}. ((a1 \mmember{} A) {}\mRightarrow{} (a2 \mmember{} A) {}\mRightarrow{} seteq(a1;a2) {}\mRightarrow{} seteq(B[a1];B[a2])))
{}\mRightarrow{} (\mforall{}g1,g2:coSet\{i:l\}.
(seteq(g1;g2) {}\mRightarrow{} (singlevalued-graph(A;a.B[a];g1) \mLeftarrow{}{}\mRightarrow{} singlevalued-graph(A;a.B[a];g2)))))
Date html generated:
2018_07_29-AM-10_05_01
Last ObjectModification:
2018_07_18-PM-04_01_17
Theory : constructive!set!theory
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