Nuprl Lemma : map_wf_combination
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[n:ℤ]. ∀[L:Combination(n;A)]. (map(f;L) ∈ Combination(n;B)) supposing Inj(A;B;f)
Proof
Definitions occuring in Statement :
combination: Combination(n;T),
map: map(f;as),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
combination: Combination(n;T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
cand: A c∧ B,
all: ∀x:A. B[x],
top: Top,
inject: Inj(A;B;f),
implies: P ⇒ Q,
squash: ↓T
Lemmas referenced :
set_wf,
list_wf,
no_repeats_wf,
equal-wf-T-base,
length_wf,
int_subtype_base,
inject_wf,
map_wf,
no_repeats_map,
subtype_rel_dep_function,
l_member_wf,
map-length,
member_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
lambdaEquality,
productEquality,
intEquality,
applyEquality,
isect_memberEquality,
because_Cache,
functionExtensionality,
functionEquality,
universeEquality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_isectElimination,
setEquality,
lambdaFormation,
independent_pairFormation,
voidElimination,
voidEquality,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B].
\mforall{}[n:\mBbbZ{}]. \mforall{}[L:Combination(n;A)]. (map(f;L) \mmember{} Combination(n;B)) supposing Inj(A;B;f)
Date html generated:
2018_05_21-PM-08_07_59
Last ObjectModification:
2017_07_26-PM-05_43_44
Theory : general
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