Nuprl Lemma : sets-arrow
∀[C:SmallCategory]. ∀[X:Top].
(cat-arrow(sets(C; X)) ~ λAa,Bb. let A,a = Aa in let B,b = Bb in {f:cat-arrow(C) A B| f(b) = a ∈ (X A)} )
Proof
Definitions occuring in Statement :
sets: sets(C; X),
psc-restriction: f(s),
functor-ob: ob(F),
cat-arrow: cat-arrow(C),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
top: Top,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
spread: spread def,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory,
spreadn: spread4,
psc-restriction: f(s),
sets: sets(C; X),
all: ∀x:A. B[x],
top: Top,
presheaf-elements: el(P),
mk-cat: mk-cat,
and: P ∧ Q,
functor-arrow: arrow(F),
pi2: snd(t)
Lemmas referenced :
cat_arrow_triple_lemma,
op-cat-arrow,
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-arrow(sets(C; X)) \msim{} \mlambda{}Aa,Bb. let A,a = Aa in let B,b = Bb in \{f:cat-arrow(C) A B| f(b) = a\} )
Date html generated:
2018_05_22-PM-09_59_17
Last ObjectModification:
2018_05_20-PM-09_42_07
Theory : presheaf!models!of!type!theory
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