Nuprl Lemma : respects-equality-sets
∀[A,T:Type]. ∀[P:T ⟶ ℙ]. ∀[Q:A ⟶ ℙ]. (respects-equality(A;T) ⇒ respects-equality({x:A| Q[x]} ;{x:T| P[x]} ))
Proof
Definitions occuring in Statement :
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
so_lambda: λ2x.t[x],
respects-equality: respects-equality(S;T),
all: ∀x:A. B[x],
squash: ↓T
Lemmas referenced :
respects-equality-set,
istype-base,
respects-equality_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
sqequalRule,
applyEquality,
Error :lambdaEquality_alt,
Error :universeIsType,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
Error :equalityIstype,
Error :setIsType,
because_Cache,
sqequalBase,
equalitySymmetry,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsType,
universeEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies
Latex:
\mforall{}[A,T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}]. \mforall{}[Q:A {}\mrightarrow{} \mBbbP{}]. (respects-equality(A;T) {}\mRightarrow{} respects-equality(\{x:A| Q[x]\} ;\{x\000C:T| P[x]\} ))
Date html generated:
2019_06_20-AM-11_13_46
Last ObjectModification:
2018_12_13-PM-00_07_02
Theory : core_2
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