Nuprl Lemma : setmem-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)))
⇒ (∀y:coSet{i:l}. ((y ∈ fun-graph(b;f)) ⇐⇒ ∃p:x:coSet{i:l} × (x ∈ b). seteq(y;(fst(p),f p)))))
Proof
Definitions occuring in Statement :
fun-graph: fun-graph(b;f),
orderedpairset: (a,b),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x]
Definitions unfolded in proof :
true: True,
squash: ↓T,
top: Top,
Wsup: Wsup(a;b),
mk-set: f"(T),
pi1: fst(t),
set-dom: set-dom(s),
pi2: snd(t),
set-item: set-item(s;x),
fun-graph: fun-graph(b;f),
mk-coset: mk-coset(T;f),
subtype_rel: A ⊆r B,
so_apply: x[s],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
true_wf,
squash_wf,
seteq_weakening,
seteq_functionality,
seteq-orderedpairs-iff,
seteqweaken_wf,
setmem-mk-coset,
subtype_rel_self,
mem-mk-set_wf2,
mk-coset_wf,
setmem-iff,
coSet_subtype,
subtype_coSet,
all_wf,
pi1_wf,
orderedpairset_wf,
seteq_wf,
coSet_wf,
exists_wf,
fun-graph_wf,
setmem_wf
Rules used in proof :
levelHypothesis,
baseClosed,
imageMemberEquality,
natural_numberEquality,
equalityTransitivity,
imageElimination,
equalitySymmetry,
hyp_replacement,
addLevel,
because_Cache,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_pairEquality,
dependent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
hypothesis_subsumption,
functionEquality,
applyEquality,
lambdaEquality,
sqequalRule,
cumulativity,
productEquality,
instantiate,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
independent_pairFormation,
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{}y:coSet\{i:l\}. ((y \mmember{} fun-graph(b;f)) \mLeftarrow{}{}\mRightarrow{} \mexists{}p:x:coSet\{i:l\} \mtimes{} (x \mmember{} b). seteq(y;(fst(p),f p)))))
Date html generated:
2018_07_29-AM-10_09_14
Last ObjectModification:
2018_07_18-PM-09_37_41
Theory : constructive!set!theory
Home
Index