Nuprl Lemma : inject_functionality
∀[A,B,C:Type]. ∀[f:B ⟶ C]. (Inj(A;C;f)) supposing (Inj(B;C;f) and strong-subtype(A;B))
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],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
strong-subtype: strong-subtype(A;B),
cand: A c∧ B,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
inject: Inj(A;B;f),
guard: {T},
label: ...$L... t
Lemmas referenced :
sq_stable__inject,
subtype_rel_dep_function,
inject_wf,
strong-subtype_wf,
strong-subtype-implies,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
lambdaEquality,
cumulativity,
independent_isectElimination,
hypothesis,
productElimination,
lambdaFormation,
because_Cache,
independent_functionElimination,
promote_hyp,
imageMemberEquality,
baseClosed,
imageElimination,
functionExtensionality,
functionEquality,
universeEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[f:B {}\mrightarrow{} C]. (Inj(A;C;f)) supposing (Inj(B;C;f) and strong-subtype(A;B))
Date html generated:
2017_04_14-AM-07_36_57
Last ObjectModification:
2017_02_27-PM-03_09_04
Theory : subtype_1
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