Nuprl Lemma : comem-graph-cosets
∀[I:Type]. ∀[E:I ⟶ I ⟶ ℙ].
∀i:I. ∀x:coSet{i:l}.
(comem{i:l}(x;graph-cosets(I;i,j.E[i;j]) i) ⇐⇒ ∃j:I. (E[i;j] ∧ (x = (graph-cosets(I;i,j.E[i;j]) j) ∈ coSet{i:l})))
Proof
Definitions occuring in Statement :
graph-cosets: graph-cosets(I;i,j.E[i; j]),
comem: comem{i:l}(x;s),
coSet: coSet{i:l},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
prop: ℙ,
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
pi2: snd(t),
pi1: fst(t),
set-dom: set-dom(s),
set-item: set-item(s;x),
comem: comem{i:l}(x;s),
graph-cosets: graph-cosets(I;i,j.E[i; j]),
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
and_wf,
pi1_wf,
graph-cosets_wf,
coSet_wf,
equal_wf,
subtype_rel_self,
exists_wf
Rules used in proof :
setElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
dependent_pairEquality,
rename,
dependent_pairFormation,
universeEquality,
functionEquality,
lambdaEquality,
because_Cache,
hypothesis,
applyEquality,
hypothesisEquality,
productEquality,
cumulativity,
isectElimination,
extract_by_obid,
introduction,
instantiate,
cut,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
sqequalRule,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[I:Type]. \mforall{}[E:I {}\mrightarrow{} I {}\mrightarrow{} \mBbbP{}].
\mforall{}i:I. \mforall{}x:coSet\{i:l\}.
(comem\{i:l\}(x;graph-cosets(I;i,j.E[i;j]) i)
\mLeftarrow{}{}\mRightarrow{} \mexists{}j:I. (E[i;j] \mwedge{} (x = (graph-cosets(I;i,j.E[i;j]) j))))
Date html generated:
2018_07_29-AM-09_50_21
Last ObjectModification:
2018_07_11-PM-10_45_58
Theory : constructive!set!theory
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