Nuprl Lemma : injection-inverse
∀[A,B:Type].  ∀f:A →⟶ B. (A 
⇒ finite-type(A) 
⇒ (∀x,y:B.  Dec(x = y ∈ B)) 
⇒ (∃g:B ⟶ A. ∀a:A. ((g (f a)) = a ∈ A)))
Proof
Definitions occuring in Statement : 
injection: A →⟶ B
, 
finite-type: finite-type(T)
, 
decidable: Dec(P)
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
implies: 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]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
finite-type: finite-type(T)
, 
exists: ∃x:A. B[x]
, 
member: t ∈ T
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
nat: ℕ
, 
injection: A →⟶ B
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
pi1: fst(t)
, 
guard: {T}
, 
inject: Inj(A;B;f)
, 
not: ¬A
, 
false: False
, 
surject: Surj(A;B;f)
Lemmas referenced : 
all_wf, 
decidable_wf, 
equal_wf, 
finite-type_wf, 
injection_wf, 
decidable__exists_int_seg, 
int_seg_wf, 
exists_wf, 
not_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
rename, 
sqequalHypSubstitution, 
productElimination, 
thin, 
cut, 
lemma_by_obid, 
isectElimination, 
hypothesisEquality, 
sqequalRule, 
lambdaEquality, 
hypothesis, 
universeEquality, 
instantiate, 
dependent_functionElimination, 
natural_numberEquality, 
setElimination, 
applyEquality, 
independent_functionElimination, 
dependent_pairFormation, 
unionEquality, 
unionElimination, 
equalityTransitivity, 
equalitySymmetry, 
cumulativity, 
because_Cache, 
voidElimination
Latex:
\mforall{}[A,B:Type].
    \mforall{}f:A  \mrightarrow{}{}\mrightarrow{}  B.  (A  {}\mRightarrow{}  finite-type(A)  {}\mRightarrow{}  (\mforall{}x,y:B.    Dec(x  =  y))  {}\mRightarrow{}  (\mexists{}g:B  {}\mrightarrow{}  A.  \mforall{}a:A.  ((g  (f  a))  =  a)))
Date html generated:
2016_05_15-PM-06_11_25
Last ObjectModification:
2015_12_27-PM-00_14_11
Theory : general
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