Nuprl Lemma : function-graph_wf
∀[A:coSet{i:l}]. ∀[B:{a:coSet{i:l}| (a ∈ A)} ⟶ coSet{i:l}]. ∀[grph:coSet{i:l}].
(function-graph{i:l}(A;a.B[a];grph) ∈ ℙ')
Proof
Definitions occuring in Statement :
function-graph: function-graph{i:l}(A;a.B[a];grph),
setmem: (x ∈ s),
coSet: coSet{i:l},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
and: P ∧ Q,
prop: ℙ,
function-graph: function-graph{i:l}(A;a.B[a];grph),
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
seteq_wf,
orderedpairset_wf,
exists_wf,
all_wf,
setmem_wf,
coSet_wf,
sigmaset_wf,
setsubset_wf
Rules used in proof :
isect_memberEquality,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
because_Cache,
dependent_set_memberEquality,
functionEquality,
instantiate,
hypothesis,
setEquality,
applyEquality,
lambdaEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cumulativity,
productEquality,
sqequalRule,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:coSet\{i:l\}]. \mforall{}[B:\{a:coSet\{i:l\}| (a \mmember{} A)\} {}\mrightarrow{} coSet\{i:l\}]. \mforall{}[grph:coSet\{i:l\}].
(function-graph\{i:l\}(A;a.B[a];grph) \mmember{} \mBbbP{}')
Date html generated:
2018_07_29-AM-10_05_15
Last ObjectModification:
2018_07_18-PM-04_39_04
Theory : constructive!set!theory
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