Nuprl Lemma : apply-wf-per
∀[A:Type]. ∀[B:type-function{i:l}(A)]. ∀[f:per-function(A;a.B[a])]. ∀[a:A]. (f[a] ∈ B[a])
Proof
Definitions occuring in Statement :
type-function: type-function{i:l}(A),
per-function: per-function(A;a.B[a]),
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
per-function: per-function(A;a.B[a]),
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
function-eq: function-eq(A;a.B[a];f;g),
squash: ↓T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
true: True
Lemmas referenced :
per-function_wf,
function-eq_wf_type_function,
and_wf,
equal_wf,
apply_wf_type-function,
member_wf,
type-function_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
pointwiseFunctionality,
sqequalRule,
pertypeElimination,
dependent_functionElimination,
equalitySymmetry,
dependent_set_memberEquality,
hypothesis,
independent_pairFormation,
equalityTransitivity,
applyLambdaEquality,
setElimination,
rename,
productElimination,
applyEquality,
lambdaEquality,
imageElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
universeEquality,
hyp_replacement
Latex:
\mforall{}[A:Type]. \mforall{}[B:type-function\{i:l\}(A)]. \mforall{}[f:per-function(A;a.B[a])]. \mforall{}[a:A]. (f[a] \mmember{} B[a])
Date html generated:
2017_04_14-AM-07_29_17
Last ObjectModification:
2017_02_27-PM-02_57_34
Theory : per!type
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