Nuprl Lemma : ab_binrel_functionality
∀[T:Type]. ∀[E,E':T ⟶ T ⟶ ℙ]. ((∀x,y:T. (E[x;y] ⇐⇒ E'[x;y])) ⇒ ((x,y:T. E[x;y]) <≡>{T} (x,y:T. E'[x;y])))
Proof
Definitions occuring in Statement :
ab_binrel: x,y:T. E[x; y],
binrel_eqv: E <≡>{T} E',
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
ab_binrel: x,y:T. E[x; y],
binrel_eqv: E <≡>{T} E',
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s]
Lemmas referenced :
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
hypothesis,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[E,E':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. (E[x;y] \mLeftarrow{}{}\mRightarrow{} E'[x;y])) {}\mRightarrow{} ((x,y:T. E[x;y]) <\mequiv{}>\{T\} (x,y:T. E'[x;y])))
Date html generated:
2016_05_15-PM-00_00_36
Last ObjectModification:
2015_12_26-PM-11_26_39
Theory : gen_algebra_1
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