Nuprl Lemma : bijection_restriction
∀k:ℕ. ∀f:ℕk ⟶ ℕk.
  Bij(ℕk;ℕk;f) 
⇒ {(f ∈ ℕk - 1 ⟶ ℕk - 1) ∧ Bij(ℕk - 1;ℕk - 1;f)} supposing (f (k - 1)) = (k - 1) ∈ ℤ supposing 0 < k
Proof
Definitions occuring in Statement : 
biject: Bij(A;B;f)
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
less_than: a < b
, 
uimplies: b supposing a
, 
guard: {T}
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
and: P ∧ Q
, 
member: t ∈ T
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
subtract: n - m
, 
natural_number: $n
, 
int: ℤ
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
nat: ℕ
, 
implies: P 
⇒ Q
, 
guard: {T}
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
prop: ℙ
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
iff: P 
⇐⇒ Q
, 
not: ¬A
, 
rev_implies: P 
⇐ Q
, 
false: False
, 
uiff: uiff(P;Q)
, 
subtract: n - m
, 
subtype_rel: A ⊆r B
, 
top: Top
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
true: True
, 
less_than: a < b
, 
exists: ∃x:A. B[x]
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
sq_type: SQType(T)
, 
biject: Bij(A;B;f)
, 
inject: Inj(A;B;f)
, 
squash: ↓T
, 
surject: Surj(A;B;f)
Lemmas referenced : 
member-less_than, 
equal_wf, 
int_seg_wf, 
subtract_wf, 
decidable__le, 
false_wf, 
not-le-2, 
less-iff-le, 
condition-implies-le, 
minus-one-mul, 
zero-add, 
minus-one-mul-top, 
nat_wf, 
minus-add, 
minus-minus, 
add-associates, 
add-swap, 
add-commutes, 
add_functionality_wrt_le, 
add-zero, 
le-add-cancel, 
decidable__lt, 
not-lt-2, 
add-mul-special, 
zero-mul, 
le-add-cancel-alt, 
lelt_wf, 
biject_wf, 
less_than_wf, 
decidable__int_equal, 
le-add-cancel2, 
set_subtype_base, 
int_subtype_base, 
two-mul, 
mul-distributes-right, 
one-mul, 
subtype_base_sq, 
not-equal-2, 
int_seg_properties, 
nat_properties, 
int_seg_subtype, 
le_antisymmetry_iff, 
less_than_transitivity1, 
le_weakening, 
less_than_irreflexivity, 
equal_functionality_wrt_subtype_rel2
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
isect_memberFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
natural_numberEquality, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
independent_isectElimination, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
independent_pairFormation, 
intEquality, 
applyEquality, 
functionExtensionality, 
because_Cache, 
dependent_set_memberEquality, 
dependent_functionElimination, 
unionElimination, 
voidElimination, 
productElimination, 
independent_functionElimination, 
addEquality, 
sqequalRule, 
lambdaEquality, 
isect_memberEquality, 
voidEquality, 
minusEquality, 
functionEquality, 
dependent_pairFormation, 
sqequalIntensionalEquality, 
promote_hyp, 
multiplyEquality, 
instantiate, 
cumulativity, 
applyLambdaEquality, 
imageMemberEquality, 
baseClosed
Latex:
\mforall{}k:\mBbbN{}.  \mforall{}f:\mBbbN{}k  {}\mrightarrow{}  \mBbbN{}k.
    Bij(\mBbbN{}k;\mBbbN{}k;f)  {}\mRightarrow{}  \{(f  \mmember{}  \mBbbN{}k  -  1  {}\mrightarrow{}  \mBbbN{}k  -  1)  \mwedge{}  Bij(\mBbbN{}k  -  1;\mBbbN{}k  -  1;f)\}  supposing  (f  (k  -  1))  =  (k  -  1) 
    supposing  0  <  k
Date html generated:
2017_04_14-AM-07_34_14
Last ObjectModification:
2017_02_27-PM-03_08_51
Theory : fun_1
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