Nuprl Lemma : biject-inverse
∀[A,B:Type]. ∀[f:A ⟶ B]. (Bij(A;B;f) ⇒ (∃g:B ⟶ A. ((∀b:B. ((f (g b)) = b ∈ B)) ∧ (∀a:A. ((g (f a)) = a ∈ A)))))
Proof
Definitions occuring in Statement :
biject: Bij(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: 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],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
prop: ℙ,
guard: {T},
biject: Bij(A;B;f),
inject: Inj(A;B;f),
squash: ↓T,
true: True,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bij_inv_wf,
biject_wf,
istype-universe,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
rename,
dependent_pairFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
universeIsType,
independent_pairFormation,
because_Cache,
productIsType,
functionIsType,
equalityIstype,
instantiate,
universeEquality,
dependent_functionElimination,
productElimination,
independent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B].
(Bij(A;B;f) {}\mRightarrow{} (\mexists{}g:B {}\mrightarrow{} A. ((\mforall{}b:B. ((f (g b)) = b)) \mwedge{} (\mforall{}a:A. ((g (f a)) = a)))))
Date html generated:
2020_05_19-PM-09_43_38
Last ObjectModification:
2020_01_04-PM-07_59_56
Theory : list_1
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