Nuprl Lemma : setimage-iff
∀x,b:coSet{i:l}.
(setimage{i:l}(x;b)
⇐⇒ ∃f:(z:coSet{i:l} × (z ∈ b)) ⟶ coSet{i:l}
((∀z1,z2:z:coSet{i:l} × (z ∈ b). (seteq(fst(z1);fst(z2)) ⇒ seteq(f z1;f z2))) ∧ seteq(x;set-image(f;b))))
Proof
Definitions occuring in Statement :
set-image: set-image(f;b),
setimage: setimage{i:l}(x;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,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
product: x:A × B[x]
Definitions unfolded in proof :
guard: {T},
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
cand: A c∧ B,
prop: ℙ,
member: t ∈ T,
exists: ∃x:A. B[x],
setimage: setimage{i:l}(x;b),
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
seteq_weakening,
setmem_functionality,
setmem-image,
co-seteq-iff,
exists_wf,
iff_wf,
setimage_wf,
set-image_wf,
pi1_wf,
seteq_wf,
setmem_wf,
coSet_wf,
all_wf
Rules used in proof :
because_Cache,
levelHypothesis,
impliesFunctionality,
allFunctionality,
addLevel,
independent_functionElimination,
dependent_functionElimination,
applyEquality,
functionEquality,
lambdaEquality,
sqequalRule,
cumulativity,
isectElimination,
extract_by_obid,
introduction,
instantiate,
cut,
productEquality,
hypothesis,
promote_hyp,
hypothesisEquality,
dependent_pairFormation,
thin,
productElimination,
sqequalHypSubstitution,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x,b:coSet\{i:l\}.
(setimage\{i:l\}(x;b)
\mLeftarrow{}{}\mRightarrow{} \mexists{}f:(z:coSet\{i:l\} \mtimes{} (z \mmember{} b)) {}\mrightarrow{} coSet\{i:l\}
((\mforall{}z1,z2:z:coSet\{i:l\} \mtimes{} (z \mmember{} b). (seteq(fst(z1);fst(z2)) {}\mRightarrow{} seteq(f z1;f z2)))
\mwedge{} seteq(x;set-image(f;b))))
Date html generated:
2018_07_29-AM-10_08_57
Last ObjectModification:
2018_07_18-PM-09_14_12
Theory : constructive!set!theory
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