Nuprl Lemma : biject-iff-inverse
∀[A,B:Type]. ∀f:A ⟶ B. (∃g:B ⟶ A. InvFuns(A;B;f;g) ⇐⇒ Bij(A;B;f))
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
inv_funs: InvFuns(A;B;f;g),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
universe: Type
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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
exists_wf,
inv_funs_wf,
biject_wf,
fun_with_inv_is_bij,
biject-inverse2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
universeEquality,
productElimination,
dependent_functionElimination,
independent_isectElimination,
independent_functionElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. (\mexists{}g:B {}\mrightarrow{} A. InvFuns(A;B;f;g) \mLeftarrow{}{}\mRightarrow{} Bij(A;B;f))
Date html generated:
2016_05_14-PM-01_53_50
Last ObjectModification:
2015_12_26-PM-05_39_55
Theory : list_1
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