Nuprl Lemma : inv_funs-iff
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[g:B ⟶ A]. (InvFuns(A;B;f;g) ⇐⇒ (∀a:A. ((g (f a)) = a ∈ A)) ∧ (∀b:B. ((f (g b)) = b ∈ B)))
Proof
Definitions occuring in Statement :
inv_funs: InvFuns(A;B;f;g),
uall: ∀[x:A]. B[x],
all: ∀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 :
inv_funs: InvFuns(A;B;f;g),
tidentity: Id{T},
identity: Id,
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
compose: f o g,
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-T-base,
compose_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
hypothesisEquality,
productEquality,
extract_by_obid,
isectElimination,
functionEquality,
cumulativity,
functionExtensionality,
applyEquality,
baseClosed,
because_Cache,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
independent_isectElimination,
independent_functionElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[g:B {}\mrightarrow{} A].
(InvFuns(A;B;f;g) \mLeftarrow{}{}\mRightarrow{} (\mforall{}a:A. ((g (f a)) = a)) \mwedge{} (\mforall{}b:B. ((f (g b)) = b)))
Date html generated:
2017_04_14-AM-07_33_16
Last ObjectModification:
2017_02_27-PM-03_07_13
Theory : fun_1
Home
Index