Nuprl Lemma : quo-lift_wf
∀A,B:Type. ∀f:A ⟶ B. ∀R:B ⟶ B ⟶ ℙ. (EquivRel(B;x,y.x R y) ⇒ (quo-lift(f) ∈ (x,y:A//(x R_f y)) ⟶ (x,y:B//(x R y))))
Proof
Definitions occuring in Statement :
quo-lift: quo-lift(f),
preima_of_rel: R_f,
equiv_rel: EquivRel(T;x,y.E[x; y]),
quotient: x,y:A//B[x; y],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
uimplies: b supposing a,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
preima_of_rel: R_f,
quo-lift: quo-lift(f)
Lemmas referenced :
preima_of_equiv_rel,
quotient_wf,
equal-wf-base,
preima_of_rel_wf,
equiv_rel_wf,
quotient-member-eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
functionExtensionality,
pointwiseFunctionalityForEquality,
isectElimination,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
independent_isectElimination,
pertypeElimination,
productElimination,
productEquality,
because_Cache,
universeEquality,
functionEquality
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{} (quo-lift(f) \mmember{} (x,y:A//(x R\_f y)) {}\mrightarrow{} (x,y:B//(x R y))))
Date html generated:
2016_10_21-AM-09_44_13
Last ObjectModification:
2016_08_08-PM-09_15_45
Theory : quot_1
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