Nuprl Lemma : biject-iff
∀[A,B:Type]. ∀f:A ⟶ B. (Bij(A;B;f) ⇐⇒ Inj(A;B;f) ∧ (∃g:B ⟶ A. ∀x:B. ((f (g x)) = x ∈ B)))
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
inject: Inj(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: 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],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
inject: Inj(A;B;f),
prop: ℙ,
biject: Bij(A;B;f),
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
surject: Surj(A;B;f),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
surject-inverse,
biject_wf,
squash_wf,
true_wf,
iff_weakening_equal,
inject_wf,
exists_wf,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
cumulativity,
applyEquality,
functionExtensionality,
productElimination,
independent_functionElimination,
dependent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productEquality,
functionEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. (Bij(A;B;f) \mLeftarrow{}{}\mRightarrow{} Inj(A;B;f) \mwedge{} (\mexists{}g:B {}\mrightarrow{} A. \mforall{}x:B. ((f (g x)) = x)))
Date html generated:
2017_04_17-AM-07_46_32
Last ObjectModification:
2017_02_27-PM-04_17_44
Theory : list_1
Home
Index