Nuprl Lemma : bij_inv_wf
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[bi:Bij(A;B;f)].
(bij_inv(bi) ∈ {g:B ⟶ A| (∀b:B. ((f (g b)) = b ∈ B)) ∧ (∀a:A. ((g (f a)) = a ∈ A))} )
Proof
Definitions occuring in Statement :
bij_inv: bij_inv(bi),
biject: Bij(A;B;f),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bij_inv: bij_inv(bi),
biject: Bij(A;B;f),
and: P ∧ Q,
pi2: snd(t),
surject: Surj(A;B;f),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
exists: ∃x:A. B[x],
cand: A c∧ B,
guard: {T},
inject: Inj(A;B;f),
pi1: fst(t)
Lemmas referenced :
exists_wf,
equal_wf,
pi1_wf,
all_wf,
biject_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
dependent_set_memberEquality,
lambdaEquality,
applyEquality,
hypothesisEquality,
extract_by_obid,
isectElimination,
cumulativity,
functionExtensionality,
hypothesis,
lambdaFormation,
dependent_pairEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
productEquality,
axiomEquality,
isect_memberEquality,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[bi:Bij(A;B;f)].
(bij\_inv(bi) \mmember{} \{g:B {}\mrightarrow{} A| (\mforall{}b:B. ((f (g b)) = b)) \mwedge{} (\mforall{}a:A. ((g (f a)) = a))\} )
Date html generated:
2017_04_17-AM-07_46_42
Last ObjectModification:
2017_02_27-PM-04_18_04
Theory : list_1
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