Nuprl Lemma : union-set-is-set-exists
∀[A,B:Type]. ∀[P:A ⟶ B ⟶ ℙ]. ⋃x:A.{y:B| P[x;y]} ≡ {y:B| ∃x:A. P[x;y]}
Proof
Definitions occuring in Statement :
ext-eq: A ≡ B,
tunion: ⋃x:A.B[x],
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
exists: ∃x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
tunion: ⋃x:A.B[x],
pi2: snd(t)
Lemmas referenced :
exists_wf,
tunion_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaEquality,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
setEquality,
applyEquality,
hypothesis,
universeEquality,
productElimination,
independent_pairEquality,
axiomEquality,
functionEquality,
cumulativity,
isect_memberEquality,
because_Cache,
imageElimination,
setElimination,
rename,
dependent_set_memberEquality,
dependent_pairFormation,
imageMemberEquality,
dependent_pairEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:A {}\mrightarrow{} B {}\mrightarrow{} \mBbbP{}]. \mcup{}x:A.\{y:B| P[x;y]\} \mequiv{} \{y:B| \mexists{}x:A. P[x;y]\}
Date html generated:
2016_05_13-PM-04_14_05
Last ObjectModification:
2016_01_14-PM-07_29_03
Theory : subtype_1
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