Nuprl Lemma : bij_imp_exists_inv
∀[A,B:Type]. ∀f:A ⟶ B. (Bij(A;B;f) ⇒ (∃g:B ⟶ A. InvFuns(A;B;f;g)))
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,
member: t ∈ T,
prop: ℙ,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
inject: Inj(A;B;f),
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
inv_funs: InvFuns(A;B;f;g),
compose: f o g,
tidentity: Id{T},
identity: Id,
guard: {T},
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
biject_wf,
ax_choice,
equal_wf,
inv_funs_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
functionEquality,
Error :inhabitedIsType,
Error :universeIsType,
universeEquality,
productElimination,
sqequalRule,
lambdaEquality,
applyEquality,
independent_functionElimination,
dependent_pairFormation,
independent_pairFormation,
functionExtensionality,
dependent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}f:A {}\mrightarrow{} B. (Bij(A;B;f) {}\mRightarrow{} (\mexists{}g:B {}\mrightarrow{} A. InvFuns(A;B;f;g)))
Date html generated:
2019_06_20-PM-00_26_32
Last ObjectModification:
2019_06_19-PM-06_20_06
Theory : fun_1
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