Nuprl Lemma : eqfun_p_subtyping
∀[T:Type]. ∀[P:T ⟶ ℙ]. ∀[eq:T ⟶ T ⟶ 𝔹]. IsEqFun({x:T| P[x]} ;eq) supposing IsEqFun(T;eq)
Proof
Definitions occuring in Statement :
eqfun_p: IsEqFun(T;eq),
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
eqfun_p: IsEqFun(T;eq),
member: t ∈ T,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
assert_wf,
equal_wf,
set_wf,
eqfun_p_wf,
bool_wf,
assert_witness,
equal_functionality_wrt_subtype_rel2,
iff_weakening_uiff,
uiff_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
setElimination,
rename,
hypothesis,
setEquality,
lambdaEquality,
sqequalRule,
universeEquality,
dependent_set_memberEquality,
because_Cache,
functionEquality,
isect_memberFormation,
isect_memberEquality,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
independent_pairFormation,
independent_isectElimination,
addLevel
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[eq:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. IsEqFun(\{x:T| P[x]\} ;eq) supposing IsEqFun(T;eq)
Date html generated:
2017_10_01-AM-08_13_21
Last ObjectModification:
2017_02_28-PM-01_57_49
Theory : sets_1
Home
Index