Nuprl Lemma : function-graph-fun-graph
∀b:coSet{i:l}. ∀f:(x:coSet{i:l} × (x ∈ b)) ⟶ coSet{i:l}.
((∀z1,z2:x:coSet{i:l} × (x ∈ b). (seteq(fst(z1);fst(z2)) ⇒ seteq(f z1;f z2)))
⇒ (∀x:coSet{i:l}. ((set-image(f;b) ⊆ x) ⇒ function-graph{i:l}(b;_.x;fun-graph(b;f)))))
Proof
Definitions occuring in Statement :
fun-graph: fun-graph(b;f),
set-image: set-image(f;b),
function-graph: function-graph{i:l}(A;a.B[a];grph),
setsubset: (a ⊆ b),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x]
Definitions unfolded in proof :
orderedpairset: (a,b),
guard: {T},
pi1: fst(t),
cand: A c∧ B,
function-graph: function-graph{i:l}(A;a.B[a];grph),
exists: ∃x:A. B[x],
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
seteq_transitivity,
seteq_inversion,
seteq_functionality,
seteq-orderedpairs-iff,
singleset_wf,
pairset_wf,
sigmaset_wf,
setmem-sigmaset,
setmem-fun-graph,
pi1_wf,
all_wf,
setsubset_wf,
fun-graph_wf,
orderedpairset_wf,
seteq_weakening,
seteq_wf,
setmem_wf,
coSet_wf,
exists_wf,
setmem-image,
set-image_wf,
setsubset-iff
Rules used in proof :
rename,
dependent_pairEquality,
impliesLevelFunctionality,
allLevelFunctionality,
levelHypothesis,
setEquality,
impliesFunctionality,
addLevel,
functionEquality,
independent_pairFormation,
because_Cache,
dependent_pairFormation,
promote_hyp,
applyEquality,
lambdaEquality,
sqequalRule,
cumulativity,
productEquality,
instantiate,
allFunctionality,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}b:coSet\{i:l\}. \mforall{}f:(x:coSet\{i:l\} \mtimes{} (x \mmember{} b)) {}\mrightarrow{} coSet\{i:l\}.
((\mforall{}z1,z2:x:coSet\{i:l\} \mtimes{} (x \mmember{} b). (seteq(fst(z1);fst(z2)) {}\mRightarrow{} seteq(f z1;f z2)))
{}\mRightarrow{} (\mforall{}x:coSet\{i:l\}. ((set-image(f;b) \msubseteq{} x) {}\mRightarrow{} function-graph\{i:l\}(b;$_{}$.x;fun-\000Cgraph(b;f)))))
Date html generated:
2018_07_29-AM-10_09_20
Last ObjectModification:
2018_07_18-PM-10_10_46
Theory : constructive!set!theory
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