Nuprl Lemma : allsetmem_functionality
∀A:coSet{i:l}. ∀P:{a:coSet{i:l}| (a ∈ A)} ⟶ ℙ. ∀B:coSet{i:l}.
(set-predicate{i:l}(A;a.P[a]) ⇒ seteq(A;B) ⇒ (∀a∈A.P[a] ⇐⇒ ∀a∈B.P[a]))
Proof
Definitions occuring in Statement :
allsetmem: ∀a∈A.P[a],
set-predicate: set-predicate{i:l}(s;a.P[a]),
setmem: (x ∈ s),
seteq: seteq(s1;s2),
coSet: coSet{i:l},
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
set-predicate: set-predicate{i:l}(s;a.P[a]),
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
seteq_wf,
set-predicate_wf,
setmem_wf,
coSet_wf,
setmem_functionality,
seteq_weakening,
seteq_inversion,
allsetmem-iff,
allsetmem_wf,
subtype_rel_self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality_alt,
cumulativity,
inhabitedIsType,
setElimination,
rename,
applyEquality,
dependent_set_memberEquality_alt,
universeEquality,
functionIsType,
setIsType,
independent_functionElimination,
productElimination,
independent_pairFormation,
promote_hyp,
instantiate
Latex:
\mforall{}A:coSet\{i:l\}. \mforall{}P:\{a:coSet\{i:l\}| (a \mmember{} A)\} {}\mrightarrow{} \mBbbP{}. \mforall{}B:coSet\{i:l\}.
(set-predicate\{i:l\}(A;a.P[a]) {}\mRightarrow{} seteq(A;B) {}\mRightarrow{} (\mforall{}a\mmember{}A.P[a] \mLeftarrow{}{}\mRightarrow{} \mforall{}a\mmember{}B.P[a]))
Date html generated:
2019_10_31-AM-06_33_33
Last ObjectModification:
2018_11_10-PM-00_34_57
Theory : constructive!set!theory
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