Nuprl Lemma : compact_functionality_wrt_surject
∀[T,S:Type]. ((∃f:T ⟶ S. Surj(T;S;f)) ⇒ compact-type(T) ⇒ compact-type(S))
Proof
Definitions occuring in Statement :
compact-type: compact-type(T),
surject: Surj(A;B;f),
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
compact-type: compact-type(T),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
compose: f o g,
guard: {T},
surject: Surj(A;B;f),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bool_wf,
compact-type_wf,
exists_wf,
surject_wf,
compose_wf,
equal-wf-T-base,
all_wf,
equal_wf,
squash_wf,
true_wf,
btrue_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
functionEquality,
cumulativity,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality,
functionExtensionality,
applyEquality,
universeEquality,
dependent_functionElimination,
unionElimination,
inlFormation,
dependent_pairFormation,
baseClosed,
inrFormation,
equalitySymmetry,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
independent_isectElimination,
independent_functionElimination,
because_Cache,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T,S:Type]. ((\mexists{}f:T {}\mrightarrow{} S. Surj(T;S;f)) {}\mRightarrow{} compact-type(T) {}\mRightarrow{} compact-type(S))
Date html generated:
2017_10_01-AM-08_29_04
Last ObjectModification:
2017_07_26-PM-04_23_48
Theory : basic
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