Nuprl Lemma : subtype-fpf-general
∀[A:Type]. ∀[P:A ⟶ ℙ]. ∀[B:A ⟶ 𝕌{j}]. (a:{a:A| P[a]} fp-> B[a] ⊆r a:A fp-> B[a])
Proof
Definitions occuring in Statement :
fpf: a:A fp-> B[a],
subtype_rel: A ⊆r B,
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
fpf: a:A fp-> B[a],
so_apply: x[s],
uimplies: b supposing a,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ
Lemmas referenced :
subtype_rel_list,
l_member-set2,
l_member-settype,
l_member_wf,
fpf_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaEquality,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairEquality,
hypothesisEquality,
applyEquality,
lemma_by_obid,
isectElimination,
setEquality,
hypothesis,
because_Cache,
independent_isectElimination,
setElimination,
rename,
functionExtensionality,
dependent_set_memberEquality,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
instantiate,
axiomEquality,
universeEquality,
isect_memberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbP{}]. \mforall{}[B:A {}\mrightarrow{} \mBbbU{}\{j\}]. (a:\{a:A| P[a]\} fp-> B[a] \msubseteq{}r a:A fp-> B[a])
Date html generated:
2018_05_21-PM-09_17_00
Last ObjectModification:
2018_02_09-AM-10_16_21
Theory : finite!partial!functions
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