Nuprl Lemma : map_functionality
∀[T,A:Type]. ∀[L1,L2:T List]. ∀[f,g:{x:T| (x ∈ L1)} ⟶ A].
(map(f;L1) = map(g;L2) ∈ (A List)) supposing ((f = g ∈ ({x:T| (x ∈ L1)} ⟶ A)) and (L1 = L2 ∈ (T List)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
map: map(f;as),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True
Lemmas referenced :
list-subtype,
list_wf,
l_member_wf,
equal_wf,
subtype_rel_self,
subtype_rel_wf,
map_wf,
squash_wf,
true_wf,
set_wf,
strong-subtype-equal-lists,
strong-subtype-set3,
strong-subtype-self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
setEquality,
cumulativity,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
functionExtensionality,
applyEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
hyp_replacement,
applyLambdaEquality,
lambdaEquality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination
Latex:
\mforall{}[T,A:Type]. \mforall{}[L1,L2:T List]. \mforall{}[f,g:\{x:T| (x \mmember{} L1)\} {}\mrightarrow{} A].
(map(f;L1) = map(g;L2)) supposing ((f = g) and (L1 = L2))
Date html generated:
2018_05_21-PM-08_36_40
Last ObjectModification:
2017_07_26-PM-06_01_10
Theory : general
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