Nuprl Lemma : sq_stable-finite-type-onto
∀[A,B:Type]. (finite-type(A) ⇒ (∀x,y:B. Dec(x = y ∈ B)) ⇒ (∀f:A ⟶ B. ∀b:B. SqStable(∃a:A. ((f a) = b ∈ B))))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
sq_stable: SqStable(P),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
false: False,
not: ¬A
Lemmas referenced :
squash_wf,
exists_wf,
equal_wf,
decidable_wf,
finite-type_wf,
istype-universe,
decidable-exists-finite-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
Error :lambdaEquality_alt,
applyEquality,
hypothesis,
Error :functionIsType,
because_Cache,
Error :inhabitedIsType,
instantiate,
universeEquality,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
imageElimination,
voidElimination
Latex:
\mforall{}[A,B:Type].
(finite-type(A) {}\mRightarrow{} (\mforall{}x,y:B. Dec(x = y)) {}\mRightarrow{} (\mforall{}f:A {}\mrightarrow{} B. \mforall{}b:B. SqStable(\mexists{}a:A. ((f a) = b))))
Date html generated:
2019_06_20-PM-01_32_28
Last ObjectModification:
2018_10_30-PM-01_36_41
Theory : list_1
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