Nuprl Lemma : ext-family-iff
∀[P:Type]. ∀[F,G:P ⟶ Type]. uiff(F ≡ G;F ⊆ G ∧ G ⊆ F)
Proof
Definitions occuring in Statement :
ext-family: F ≡ G,
sub-family: F ⊆ G,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
sub-family: F ⊆ G,
ext-family: F ≡ G,
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
all_wf,
and_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
because_Cache,
independent_pairEquality,
lambdaEquality,
axiomEquality,
lemma_by_obid,
isectElimination,
applyEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[P:Type]. \mforall{}[F,G:P {}\mrightarrow{} Type]. uiff(F \mequiv{} G;F \msubseteq{} G \mwedge{} G \msubseteq{} F)
Date html generated:
2016_05_14-AM-06_12_13
Last ObjectModification:
2015_12_26-PM-00_06_15
Theory : co-recursion
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