Nuprl Lemma : fun_with_inv_is_bij2
∀[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],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
fun_with_inv_is_bij,
exists_wf,
inv_funs_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
hypothesis,
functionEquality,
sqequalRule,
lambdaEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. ((\mexists{}g:B {}\mrightarrow{} A. InvFuns(A;B;f;g)) {}\mRightarrow{} Bij(A;B;f))
Date html generated:
2016_05_15-PM-03_21_54
Last ObjectModification:
2015_12_27-PM-01_04_33
Theory : general
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