Nuprl Lemma : trans_functionality_wrt_iff
∀[T:Type]. ∀[R,R':T ⟶ T ⟶ ℙ]. ((∀x,y:T. (R[x;y] ⇐⇒ R'[x;y])) ⇒ (Trans(T;y,x.R[x;y]) ⇐⇒ Trans(T;y,x.R'[x;y])))
Proof
Definitions occuring in Statement :
trans: Trans(T;x,y.E[x; y]),
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,
trans: Trans(T;x,y.E[x; y]),
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
trans_wf,
subtype_rel_self,
all_wf,
iff_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
productElimination,
independent_functionElimination,
dependent_functionElimination,
cumulativity,
instantiate,
universeEquality,
functionEquality,
Error :inhabitedIsType,
Error :functionIsType,
Error :universeIsType
Latex:
\mforall{}[T:Type]. \mforall{}[R,R':T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. (R[x;y] \mLeftarrow{}{}\mRightarrow{} R'[x;y])) {}\mRightarrow{} (Trans(T;y,x.R[x;y]) \mLeftarrow{}{}\mRightarrow{} Trans(T;y,x.R'[x;y])))
Date html generated:
2019_06_20-PM-00_28_46
Last ObjectModification:
2018_09_26-AM-11_46_35
Theory : rel_1
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