Nuprl Lemma : inject-subtype
∀[C,A:Type]. ∀[B:Type]. ∀[f:A ⟶ B]. Inj(C;B;f) supposing Inj(A;B;f) supposing strong-subtype(C;A)
Proof
Definitions occuring in Statement :
strong-subtype: strong-subtype(A;B),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
inject: Inj(A;B;f),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
implies: P ⇒ Q,
label: ...$L... t,
guard: {T},
prop: ℙ
Lemmas referenced :
strong-subtype-eq1,
inject_wf,
strong-subtype_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
lambdaFormation_alt,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
productElimination,
sqequalRule,
independent_functionElimination,
extract_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
equalityIstype,
universeIsType,
lambdaEquality_alt,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
instantiate,
universeEquality
Latex:
\mforall{}[C,A:Type]. \mforall{}[B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. Inj(C;B;f) supposing Inj(A;B;f) supposing strong-subtype(C;A)
Date html generated:
2020_05_19-PM-09_36_21
Last ObjectModification:
2020_01_25-PM-08_00_58
Theory : subtype_1
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