Nuprl Lemma : setmem-mkset
∀T:Type. ∀f:T ⟶ Set{i:l}. ∀x:Set{i:l}. ((x ∈ {f[t] | t ∈ T}) ⇐⇒ ∃t:T. seteq(x;f[t]))
Proof
Definitions occuring in Statement :
setmem: (x ∈ s),
seteq: seteq(s1;s2),
mkset: {f[t] | t ∈ T},
Set: Set{i:l},
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
mkset: {f[t] | t ∈ T},
set-item: set-item(s;x),
set-dom: set-dom(s),
pi1: fst(t),
pi2: snd(t),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x]
Lemmas referenced :
exists_wf,
seteq_wf,
setmem-iff,
mkset_wf,
setmem_wf,
iff_wf,
set-dom_wf,
set-item_wf,
Set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
because_Cache,
addLevel,
productElimination,
impliesFunctionality,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}T:Type. \mforall{}f:T {}\mrightarrow{} Set\{i:l\}. \mforall{}x:Set\{i:l\}. ((x \mmember{} \{f[t] | t \mmember{} T\}) \mLeftarrow{}{}\mRightarrow{} \mexists{}t:T. seteq(x;f[t]))
Date html generated:
2018_05_22-PM-09_49_32
Last ObjectModification:
2018_05_16-PM-01_31_29
Theory : constructive!set!theory
Home
Index