Nuprl Lemma : sets-comp
∀[C:SmallCategory]. ∀[X:Top]. (cat-comp(sets(C; X)) ~ λAa,Bb,Dd,f,g. (cat-comp(C) (fst(Aa)) (fst(Bb)) (fst(Dd)) f g))
Proof
Definitions occuring in Statement :
sets: sets(C; X),
cat-comp: cat-comp(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
top: Top,
pi1: fst(t),
apply: f a,
lambda: λx.A[x],
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory,
spreadn: spread4,
sets: sets(C; X),
all: ∀x:A. B[x],
top: Top,
presheaf-elements: el(P),
mk-cat: mk-cat,
and: P ∧ Q
Lemmas referenced :
cat_comp_tuple_lemma,
op-cat-comp,
top_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
isectElimination,
sqequalAxiom,
because_Cache
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[X:Top].
(cat-comp(sets(C; X)) \msim{} \mlambda{}Aa,Bb,Dd,f,g. (cat-comp(C) (fst(Aa)) (fst(Bb)) (fst(Dd)) f g))
Date html generated:
2018_05_22-PM-09_59_19
Last ObjectModification:
2018_05_20-PM-09_42_14
Theory : presheaf!models!of!type!theory
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