Nuprl Lemma : between-fun-connected
∀[T:Type]. ∀f:T ⟶ T. (retraction(T;f) 
⇒ (∀y,x:T.  (y is f*(x) 
⇒ f x is f*(y) 
⇒ ((y = x ∈ T) ∨ (y = (f x) ∈ T)))))
Proof
Definitions occuring in Statement : 
retraction: retraction(T;f)
, 
fun-connected: y is f*(x)
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
or: 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
, 
member: t ∈ T
, 
so_lambda: λ2x y.t[x; y]
, 
prop: ℙ
, 
so_apply: x[s1;s2]
, 
or: P ∨ Q
, 
uimplies: b supposing a
, 
not: ¬A
, 
false: False
, 
guard: {T}
, 
squash: ↓T
, 
true: True
, 
subtype_rel: A ⊆r B
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
rev_implies: P 
⇐ Q
, 
retraction: retraction(T;f)
, 
exists: ∃x:A. B[x]
, 
fun-connected: y is f*(x)
, 
less_than: a < b
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
Lemmas referenced : 
fun-connected-induction, 
fun-connected_wf, 
or_wf, 
equal_wf, 
not_wf, 
retraction_wf, 
fun-connected_transitivity, 
fun-connected_weakening, 
strict-fun-connected-step, 
squash_wf, 
true_wf, 
iff_weakening_equal, 
and_wf, 
retraction-fun-path, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformless_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
dependent_functionElimination, 
sqequalRule, 
lambdaEquality, 
functionEquality, 
cumulativity, 
functionExtensionality, 
applyEquality, 
hypothesis, 
independent_functionElimination, 
inlFormation, 
because_Cache, 
axiomEquality, 
rename, 
voidElimination, 
universeEquality, 
hyp_replacement, 
equalitySymmetry, 
applyLambdaEquality, 
equalityTransitivity, 
independent_isectElimination, 
imageElimination, 
natural_numberEquality, 
imageMemberEquality, 
baseClosed, 
productElimination, 
unionElimination, 
inrFormation, 
dependent_set_memberEquality, 
independent_pairFormation, 
setElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidEquality, 
computeAll
Latex:
\mforall{}[T:Type]
    \mforall{}f:T  {}\mrightarrow{}  T.  (retraction(T;f)  {}\mRightarrow{}  (\mforall{}y,x:T.    (y  is  f*(x)  {}\mRightarrow{}  f  x  is  f*(y)  {}\mRightarrow{}  ((y  =  x)  \mvee{}  (y  =  (f  x))))))
Date html generated:
2018_05_21-PM-07_48_32
Last ObjectModification:
2017_07_26-PM-05_26_18
Theory : general
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