Nuprl Lemma : preima_of_equiv_rel
∀A,B:Type. ∀f:A ⟶ B. ∀R:B ⟶ B ⟶ ℙ. (EquivRel(B;x,y.x R y) ⇒ EquivRel(A;x,y.x R_f y))
Proof
Definitions occuring in Statement :
preima_of_rel: R_f,
equiv_rel: EquivRel(T;x,y.E[x; y]),
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
preima_of_rel: R_f,
equiv_rel: EquivRel(T;x,y.E[x; y]),
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
guard: {T}
Lemmas referenced :
subtype_rel_self,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
Error :universeIsType,
hypothesisEquality,
independent_pairFormation,
hypothesis,
applyEquality,
instantiate,
introduction,
extract_by_obid,
isectElimination,
universeEquality,
because_Cache,
Error :productIsType,
Error :functionIsType,
Error :inhabitedIsType,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}A,B:Type. \mforall{}f:A {}\mrightarrow{} B. \mforall{}R:B {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}. (EquivRel(B;x,y.x R y) {}\mRightarrow{} EquivRel(A;x,y.x R\_f y))
Date html generated:
2019_06_20-PM-00_33_11
Last ObjectModification:
2019_01_17-PM-00_44_38
Theory : quot_1
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