Nuprl Lemma : functor-cat_wf
∀[C1,C2:SmallCategory]. (FUN(C1;C2) ∈ SmallCategory)
Proof
Definitions occuring in Statement :
functor-cat: FUN(C1;C2),
small-category: SmallCategory,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
functor-cat: FUN(C1;C2),
small-category: SmallCategory,
subtype_rel: A ⊆r B,
spreadn: spread4,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
nat-trans: nat-trans(C;D;F;G),
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
cat-functor_wf,
nat-trans_wf,
identity-trans_wf,
trans-comp_wf,
equal_wf,
squash_wf,
true_wf,
cat-arrow_wf,
functor-ob_wf,
nat-trans-equation,
cat-comp_wf,
functor-arrow_wf,
iff_weakening_equal,
cat-ob_wf,
trans-id-property,
all_wf,
trans-comp-assoc,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
dependent_pairEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
lambdaEquality,
because_Cache,
independent_pairEquality,
applyEquality,
sqequalRule,
productEquality,
functionEquality,
functionExtensionality,
cumulativity,
universeEquality,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
independent_pairFormation,
axiomEquality,
isect_memberEquality
Latex:
\mforall{}[C1,C2:SmallCategory]. (FUN(C1;C2) \mmember{} SmallCategory)
Date html generated:
2020_05_20-AM-07_51_54
Last ObjectModification:
2017_07_28-AM-09_19_26
Theory : small!categories
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