Nuprl Lemma : CCC-surjection
∀[A,B:Type]. ((∃f:A ⟶ B. Surj(A;B;f)) ⇒ CCC(A) ⇒ CCC(B))
Proof
Definitions occuring in Statement :
contra-cc: CCC(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 :
prop: ℙ,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
surject: Surj(A;B;f),
compose: f o g,
member: t ∈ T,
all: ∀x:A. B[x],
contra-cc: CCC(T),
exists: ∃x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
istype-universe,
surject_wf,
contra-cc_wf,
subtype_rel_self,
iff_weakening_equal,
nat_wf,
compose_wf,
istype-nat
Rules used in proof :
Error :inhabitedIsType,
universeEquality,
instantiate,
Error :productIsType,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
Error :dependent_pairFormation_alt,
because_Cache,
Error :functionIsType,
isectElimination,
independent_functionElimination,
sqequalRule,
hypothesis,
extract_by_obid,
introduction,
cut,
Error :universeIsType,
hypothesisEquality,
applyEquality,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
productElimination,
sqequalHypSubstitution,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A,B:Type]. ((\mexists{}f:A {}\mrightarrow{} B. Surj(A;B;f)) {}\mRightarrow{} CCC(A) {}\mRightarrow{} CCC(B))
Date html generated:
2019_06_20-PM-03_01_01
Last ObjectModification:
2019_06_12-PM-08_57_08
Theory : continuity
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