Nuprl Lemma : biject-inverse2
∀[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,
exists: ∃x:A. B[x],
and: P ∧ Q,
inv_funs: InvFuns(A;B;f;g),
tidentity: Id{T},
identity: Id,
compose: f o g,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
biject-inverse,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
inv_funs_wf,
biject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
dependent_pairFormation,
independent_pairFormation,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
functionExtensionality,
cumulativity,
functionEquality
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:
2017_04_17-AM-07_47_04
Last ObjectModification:
2017_02_27-PM-04_17_58
Theory : list_1
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